Calculus Examples

$y = \ln (\frac{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}}{\sqrt{4 x + 1}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 x + 1}$ as ${(4 x + 1)}^{\frac{1}{2}}$ .

$y = \ln (\frac{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}}{{(4 x + 1)}^{\frac{1}{2}}})$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\ln (\frac{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}}{{(4 x + 1)}^{\frac{1}{2}}}))$

Step 3

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 4

Differentiate the right side of the equation.

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Step 4.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = \frac{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}}{{(4 x + 1)}^{\frac{1}{2}}}$ .

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Step 4.1.1

To apply the Chain Rule, set $u_{1}$ as $\frac{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}}{{(4 x + 1)}^{\frac{1}{2}}}$ .

$\frac{d}{d u_{1}} [\ln (u_{1})] \frac{d}{d x} [\frac{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}}{{(4 x + 1)}^{\frac{1}{2}}}]$

Step 4.1.2

The derivative of $\ln (u_{1})$ with respect to $u_{1}$ is $\frac{1}{u_{1}}$ .

$\frac{1}{u_{1}} \frac{d}{d x} [\frac{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}}{{(4 x + 1)}^{\frac{1}{2}}}]$

Step 4.1.3

Replace all occurrences of $u_{1}$ with $\frac{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}}{{(4 x + 1)}^{\frac{1}{2}}}$ .

$\frac{1}{\frac{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}}{{(4 x + 1)}^{\frac{1}{2}}}} \frac{d}{d x} [\frac{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}}{{(4 x + 1)}^{\frac{1}{2}}}]$

Step 4.2

Multiply by the reciprocal of the fraction to divide by $\frac{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}}{{(4 x + 1)}^{\frac{1}{2}}}$ .

$1 \frac{{(4 x + 1)}^{\frac{1}{2}}}{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}} \frac{d}{d x} [\frac{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}}{{(4 x + 1)}^{\frac{1}{2}}}]$

Step 4.3

Multiply $\frac{{(4 x + 1)}^{\frac{1}{2}}}{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}}$ by $1$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}}}{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}} \frac{d}{d x} [\frac{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}}{{(4 x + 1)}^{\frac{1}{2}}}]$

Step 4.4

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}$ and $g (x) = {(4 x + 1)}^{\frac{1}{2}}$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}}}{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}} \cdot \frac{{(4 x + 1)}^{\frac{1}{2}} \frac{d}{d x} [e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}] - e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2} \frac{d}{d x} [{(4 x + 1)}^{\frac{1}{2}}]}{{({(4 x + 1)}^{\frac{1}{2}})}^{2}}$

Step 4.5

Multiply the exponents in ${({(4 x + 1)}^{\frac{1}{2}})}^{2}$ .

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Step 4.5.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

Step 4.5.2

Cancel the common factor of $2$ .

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Step 4.5.2.1

Cancel the common factor.

Step 4.5.2.2

Rewrite the expression.

Step 4.6

Simplify.

Step 4.7

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = e^{x^{2}} (2 x + 1)$ and $g (x) = {(3 x + 1)}^{2}$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}}}{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}} \cdot \frac{{(4 x + 1)}^{\frac{1}{2}} (e^{x^{2}} (2 x + 1) \frac{d}{d x} [{(3 x + 1)}^{2}] + {(3 x + 1)}^{2} \frac{d}{d x} [e^{x^{2}} (2 x + 1)]) - e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2} \frac{d}{d x} [{(4 x + 1)}^{\frac{1}{2}}]}{4 x + 1}$

Step 4.8

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = 3 x + 1$ .

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Step 4.8.1

To apply the Chain Rule, set $u_{2}$ as $3 x + 1$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}}}{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}} \cdot \frac{{(4 x + 1)}^{\frac{1}{2}} (e^{x^{2}} (2 x + 1) (\frac{d}{d u_{2}} [{u_{2}}^{2}] \frac{d}{d x} [3 x + 1]) + {(3 x + 1)}^{2} \frac{d}{d x} [e^{x^{2}} (2 x + 1)]) - e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2} \frac{d}{d x} [{(4 x + 1)}^{\frac{1}{2}}]}{4 x + 1}$

Step 4.8.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 2$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}}}{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}} \cdot \frac{{(4 x + 1)}^{\frac{1}{2}} (e^{x^{2}} (2 x + 1) (2 u_{2} \frac{d}{d x} [3 x + 1]) + {(3 x + 1)}^{2} \frac{d}{d x} [e^{x^{2}} (2 x + 1)]) - e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2} \frac{d}{d x} [{(4 x + 1)}^{\frac{1}{2}}]}{4 x + 1}$

Step 4.8.3

Replace all occurrences of $u_{2}$ with $3 x + 1$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}}}{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}} \cdot \frac{{(4 x + 1)}^{\frac{1}{2}} (e^{x^{2}} (2 x + 1) (2 (3 x + 1) \frac{d}{d x} [3 x + 1]) + {(3 x + 1)}^{2} \frac{d}{d x} [e^{x^{2}} (2 x + 1)]) - e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2} \frac{d}{d x} [{(4 x + 1)}^{\frac{1}{2}}]}{4 x + 1}$

Step 4.9

Differentiate.

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Step 4.9.1

By the Sum Rule, the derivative of $3 x + 1$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [1]$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}}}{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}} \cdot \frac{{(4 x + 1)}^{\frac{1}{2}} (e^{x^{2}} (2 x + 1) (2 (3 x + 1) (\frac{d}{d x} [3 x] + \frac{d}{d x} [1])) + {(3 x + 1)}^{2} \frac{d}{d x} [e^{x^{2}} (2 x + 1)]) - e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2} \frac{d}{d x} [{(4 x + 1)}^{\frac{1}{2}}]}{4 x + 1}$

Step 4.9.2

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}}}{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}} \cdot \frac{{(4 x + 1)}^{\frac{1}{2}} (e^{x^{2}} (2 x + 1) (2 (3 x + 1) (3 \frac{d}{d x} [x] + \frac{d}{d x} [1])) + {(3 x + 1)}^{2} \frac{d}{d x} [e^{x^{2}} (2 x + 1)]) - e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2} \frac{d}{d x} [{(4 x + 1)}^{\frac{1}{2}}]}{4 x + 1}$

Step 4.9.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}}}{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}} \cdot \frac{{(4 x + 1)}^{\frac{1}{2}} (e^{x^{2}} (2 x + 1) (2 (3 x + 1) (3 \cdot 1 + \frac{d}{d x} [1])) + {(3 x + 1)}^{2} \frac{d}{d x} [e^{x^{2}} (2 x + 1)]) - e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2} \frac{d}{d x} [{(4 x + 1)}^{\frac{1}{2}}]}{4 x + 1}$

Step 4.9.4

Multiply $3$ by $1$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}}}{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}} \cdot \frac{{(4 x + 1)}^{\frac{1}{2}} (e^{x^{2}} (2 x + 1) (2 (3 x + 1) (3 + \frac{d}{d x} [1])) + {(3 x + 1)}^{2} \frac{d}{d x} [e^{x^{2}} (2 x + 1)]) - e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2} \frac{d}{d x} [{(4 x + 1)}^{\frac{1}{2}}]}{4 x + 1}$

Step 4.9.5

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}}}{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}} \cdot \frac{{(4 x + 1)}^{\frac{1}{2}} (e^{x^{2}} (2 x + 1) (2 (3 x + 1) (3 + 0)) + {(3 x + 1)}^{2} \frac{d}{d x} [e^{x^{2}} (2 x + 1)]) - e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2} \frac{d}{d x} [{(4 x + 1)}^{\frac{1}{2}}]}{4 x + 1}$

Step 4.9.6

Simplify the expression.

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Step 4.9.6.1

Add $3$ and $0$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}}}{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}} \cdot \frac{{(4 x + 1)}^{\frac{1}{2}} (e^{x^{2}} (2 x + 1) (2 (3 x + 1) \cdot 3) + {(3 x + 1)}^{2} \frac{d}{d x} [e^{x^{2}} (2 x + 1)]) - e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2} \frac{d}{d x} [{(4 x + 1)}^{\frac{1}{2}}]}{4 x + 1}$

Step 4.9.6.2

Multiply $3$ by $2$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}}}{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}} \cdot \frac{{(4 x + 1)}^{\frac{1}{2}} (e^{x^{2}} (2 x + 1) (6 (3 x + 1)) + {(3 x + 1)}^{2} \frac{d}{d x} [e^{x^{2}} (2 x + 1)]) - e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2} \frac{d}{d x} [{(4 x + 1)}^{\frac{1}{2}}]}{4 x + 1}$

Step 4.10

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = e^{x^{2}}$ and $g (x) = 2 x + 1$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}}}{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}} \cdot \frac{{(4 x + 1)}^{\frac{1}{2}} (e^{x^{2}} (2 x + 1) (6 (3 x + 1)) + {(3 x + 1)}^{2} (e^{x^{2}} \frac{d}{d x} [2 x + 1] + (2 x + 1) \frac{d}{d x} [e^{x^{2}}])) - e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2} \frac{d}{d x} [{(4 x + 1)}^{\frac{1}{2}}]}{4 x + 1}$

Step 4.11

Differentiate.

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Step 4.11.1

By the Sum Rule, the derivative of $2 x + 1$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [1]$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}}}{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}} \cdot \frac{{(4 x + 1)}^{\frac{1}{2}} (e^{x^{2}} (2 x + 1) (6 (3 x + 1)) + {(3 x + 1)}^{2} (e^{x^{2}} (\frac{d}{d x} [2 x] + \frac{d}{d x} [1]) + (2 x + 1) \frac{d}{d x} [e^{x^{2}}])) - e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2} \frac{d}{d x} [{(4 x + 1)}^{\frac{1}{2}}]}{4 x + 1}$

Step 4.11.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}}}{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}} \cdot \frac{{(4 x + 1)}^{\frac{1}{2}} (e^{x^{2}} (2 x + 1) (6 (3 x + 1)) + {(3 x + 1)}^{2} (e^{x^{2}} (2 \frac{d}{d x} [x] + \frac{d}{d x} [1]) + (2 x + 1) \frac{d}{d x} [e^{x^{2}}])) - e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2} \frac{d}{d x} [{(4 x + 1)}^{\frac{1}{2}}]}{4 x + 1}$

Step 4.11.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}}}{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}} \cdot \frac{{(4 x + 1)}^{\frac{1}{2}} (e^{x^{2}} (2 x + 1) (6 (3 x + 1)) + {(3 x + 1)}^{2} (e^{x^{2}} (2 \cdot 1 + \frac{d}{d x} [1]) + (2 x + 1) \frac{d}{d x} [e^{x^{2}}])) - e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2} \frac{d}{d x} [{(4 x + 1)}^{\frac{1}{2}}]}{4 x + 1}$

Step 4.11.4

Multiply $2$ by $1$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}}}{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}} \cdot \frac{{(4 x + 1)}^{\frac{1}{2}} (e^{x^{2}} (2 x + 1) (6 (3 x + 1)) + {(3 x + 1)}^{2} (e^{x^{2}} (2 + \frac{d}{d x} [1]) + (2 x + 1) \frac{d}{d x} [e^{x^{2}}])) - e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2} \frac{d}{d x} [{(4 x + 1)}^{\frac{1}{2}}]}{4 x + 1}$

Step 4.11.5

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}}}{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}} \cdot \frac{{(4 x + 1)}^{\frac{1}{2}} (e^{x^{2}} (2 x + 1) (6 (3 x + 1)) + {(3 x + 1)}^{2} (e^{x^{2}} (2 + 0) + (2 x + 1) \frac{d}{d x} [e^{x^{2}}])) - e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2} \frac{d}{d x} [{(4 x + 1)}^{\frac{1}{2}}]}{4 x + 1}$

Step 4.11.6

Simplify the expression.

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Step 4.11.6.1

Add $2$ and $0$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}}}{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}} \cdot \frac{{(4 x + 1)}^{\frac{1}{2}} (e^{x^{2}} (2 x + 1) (6 (3 x + 1)) + {(3 x + 1)}^{2} (e^{x^{2}} \cdot 2 + (2 x + 1) \frac{d}{d x} [e^{x^{2}}])) - e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2} \frac{d}{d x} [{(4 x + 1)}^{\frac{1}{2}}]}{4 x + 1}$

Step 4.11.6.2

Move $2$ to the left of $e^{x^{2}}$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}}}{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}} \cdot \frac{{(4 x + 1)}^{\frac{1}{2}} (e^{x^{2}} (2 x + 1) (6 (3 x + 1)) + {(3 x + 1)}^{2} (2 \cdot e^{x^{2}} + (2 x + 1) \frac{d}{d x} [e^{x^{2}}])) - e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2} \frac{d}{d x} [{(4 x + 1)}^{\frac{1}{2}}]}{4 x + 1}$

Step 4.12

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = x^{2}$ .

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Step 4.12.1

To apply the Chain Rule, set $u_{3}$ as $x^{2}$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}}}{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}} \cdot \frac{{(4 x + 1)}^{\frac{1}{2}} (e^{x^{2}} (2 x + 1) (6 (3 x + 1)) + {(3 x + 1)}^{2} (2 e^{x^{2}} + (2 x + 1) (\frac{d}{d u_{3}} [e^{u_{3}}] \frac{d}{d x} [x^{2}]))) - e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2} \frac{d}{d x} [{(4 x + 1)}^{\frac{1}{2}}]}{4 x + 1}$

Step 4.12.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{3}} [a^{u_{3}}]$ is $a^{u_{3}} \ln (a)$ where $a$ = $e$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}}}{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}} \cdot \frac{{(4 x + 1)}^{\frac{1}{2}} (e^{x^{2}} (2 x + 1) (6 (3 x + 1)) + {(3 x + 1)}^{2} (2 e^{x^{2}} + (2 x + 1) (e^{u_{3}} \frac{d}{d x} [x^{2}]))) - e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2} \frac{d}{d x} [{(4 x + 1)}^{\frac{1}{2}}]}{4 x + 1}$

Step 4.12.3

Replace all occurrences of $u_{3}$ with $x^{2}$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}}}{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}} \cdot \frac{{(4 x + 1)}^{\frac{1}{2}} (e^{x^{2}} (2 x + 1) (6 (3 x + 1)) + {(3 x + 1)}^{2} (2 e^{x^{2}} + (2 x + 1) (e^{x^{2}} \frac{d}{d x} [x^{2}]))) - e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2} \frac{d}{d x} [{(4 x + 1)}^{\frac{1}{2}}]}{4 x + 1}$

Step 4.13

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}}}{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}} \cdot \frac{{(4 x + 1)}^{\frac{1}{2}} (e^{x^{2}} (2 x + 1) (6 (3 x + 1)) + {(3 x + 1)}^{2} (2 e^{x^{2}} + (2 x + 1) e^{x^{2}} (2 x))) - e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2} \frac{d}{d x} [{(4 x + 1)}^{\frac{1}{2}}]}{4 x + 1}$

Step 4.14

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = 4 x + 1$ .

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Step 4.14.1

To apply the Chain Rule, set $u_{4}$ as $4 x + 1$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}}}{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}} \cdot \frac{{(4 x + 1)}^{\frac{1}{2}} (e^{x^{2}} (2 x + 1) (6 (3 x + 1)) + {(3 x + 1)}^{2} (2 e^{x^{2}} + (2 x + 1) e^{x^{2}} (2 x))) - e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2} (\frac{d}{d u_{4}} [{u_{4}}^{\frac{1}{2}}] \frac{d}{d x} [4 x + 1])}{4 x + 1}$

Step 4.14.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{4}} [{u_{4}}^{n}]$ is $n {u_{4}}^{n - 1}$ where $n = \frac{1}{2}$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}}}{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}} \cdot \frac{{(4 x + 1)}^{\frac{1}{2}} (e^{x^{2}} (2 x + 1) (6 (3 x + 1)) + {(3 x + 1)}^{2} (2 e^{x^{2}} + (2 x + 1) e^{x^{2}} (2 x))) - e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2} (\frac{1}{2} {u_{4}}^{\frac{1}{2} - 1} \frac{d}{d x} [4 x + 1])}{4 x + 1}$

Step 4.14.3

Replace all occurrences of $u_{4}$ with $4 x + 1$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}}}{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}} \cdot \frac{{(4 x + 1)}^{\frac{1}{2}} (e^{x^{2}} (2 x + 1) (6 (3 x + 1)) + {(3 x + 1)}^{2} (2 e^{x^{2}} + (2 x + 1) e^{x^{2}} (2 x))) - e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2} (\frac{1}{2} {(4 x + 1)}^{\frac{1}{2} - 1} \frac{d}{d x} [4 x + 1])}{4 x + 1}$

Step 4.15

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}}}{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}} \cdot \frac{{(4 x + 1)}^{\frac{1}{2}} (e^{x^{2}} (2 x + 1) (6 (3 x + 1)) + {(3 x + 1)}^{2} (2 e^{x^{2}} + (2 x + 1) e^{x^{2}} (2 x))) - e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2} (\frac{1}{2} {(4 x + 1)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [4 x + 1])}{4 x + 1}$

Step 4.16

Combine $- 1$ and $\frac{2}{2}$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}}}{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}} \cdot \frac{{(4 x + 1)}^{\frac{1}{2}} (e^{x^{2}} (2 x + 1) (6 (3 x + 1)) + {(3 x + 1)}^{2} (2 e^{x^{2}} + (2 x + 1) e^{x^{2}} (2 x))) - e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2} (\frac{1}{2} {(4 x + 1)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [4 x + 1])}{4 x + 1}$

Step 4.17

Combine the numerators over the common denominator.

$\frac{{(4 x + 1)}^{\frac{1}{2}}}{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}} \cdot \frac{{(4 x + 1)}^{\frac{1}{2}} (e^{x^{2}} (2 x + 1) (6 (3 x + 1)) + {(3 x + 1)}^{2} (2 e^{x^{2}} + (2 x + 1) e^{x^{2}} (2 x))) - e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2} (\frac{1}{2} {(4 x + 1)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [4 x + 1])}{4 x + 1}$

Step 4.18

Simplify the numerator.

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Step 4.18.1

Multiply $- 1$ by $2$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}}}{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}} \cdot \frac{{(4 x + 1)}^{\frac{1}{2}} (e^{x^{2}} (2 x + 1) (6 (3 x + 1)) + {(3 x + 1)}^{2} (2 e^{x^{2}} + (2 x + 1) e^{x^{2}} (2 x))) - e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2} (\frac{1}{2} {(4 x + 1)}^{\frac{1 - 2}{2}} \frac{d}{d x} [4 x + 1])}{4 x + 1}$

Step 4.18.2

Subtract $2$ from $1$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}}}{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}} \cdot \frac{{(4 x + 1)}^{\frac{1}{2}} (e^{x^{2}} (2 x + 1) (6 (3 x + 1)) + {(3 x + 1)}^{2} (2 e^{x^{2}} + (2 x + 1) e^{x^{2}} (2 x))) - e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2} (\frac{1}{2} {(4 x + 1)}^{\frac{- 1}{2}} \frac{d}{d x} [4 x + 1])}{4 x + 1}$

Step 4.19

Combine fractions.

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Step 4.19.1

Move the negative in front of the fraction.

$\frac{{(4 x + 1)}^{\frac{1}{2}}}{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}} \cdot \frac{{(4 x + 1)}^{\frac{1}{2}} (e^{x^{2}} (2 x + 1) (6 (3 x + 1)) + {(3 x + 1)}^{2} (2 e^{x^{2}} + (2 x + 1) e^{x^{2}} (2 x))) - e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2} (\frac{1}{2} {(4 x + 1)}^{- \frac{1}{2}} \frac{d}{d x} [4 x + 1])}{4 x + 1}$

Step 4.19.2

Combine $\frac{1}{2}$ and ${(4 x + 1)}^{- \frac{1}{2}}$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}}}{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}} \cdot \frac{{(4 x + 1)}^{\frac{1}{2}} (e^{x^{2}} (2 x + 1) (6 (3 x + 1)) + {(3 x + 1)}^{2} (2 e^{x^{2}} + (2 x + 1) e^{x^{2}} (2 x))) - e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2} (\frac{{(4 x + 1)}^{- \frac{1}{2}}}{2} \frac{d}{d x} [4 x + 1])}{4 x + 1}$

Step 4.19.3

Move ${(4 x + 1)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}}}{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}} \cdot \frac{{(4 x + 1)}^{\frac{1}{2}} (e^{x^{2}} (2 x + 1) (6 (3 x + 1)) + {(3 x + 1)}^{2} (2 e^{x^{2}} + (2 x + 1) e^{x^{2}} (2 x))) - e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2} (\frac{1}{2 {(4 x + 1)}^{\frac{1}{2}}} \frac{d}{d x} [4 x + 1])}{4 x + 1}$

Step 4.19.4

Combine $\frac{1}{2 {(4 x + 1)}^{\frac{1}{2}}}$ and $e^{x^{2}}$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}}}{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}} \cdot \frac{{(4 x + 1)}^{\frac{1}{2}} (e^{x^{2}} (2 x + 1) (6 (3 x + 1)) + {(3 x + 1)}^{2} (2 e^{x^{2}} + (2 x + 1) e^{x^{2}} (2 x))) - \frac{e^{x^{2}}}{2 {(4 x + 1)}^{\frac{1}{2}}} (2 x + 1) {(3 x + 1)}^{2} \frac{d}{d x} [4 x + 1]}{4 x + 1}$

Step 4.19.5

Combine ${(3 x + 1)}^{2}$ and $\frac{e^{x^{2}}}{2 {(4 x + 1)}^{\frac{1}{2}}}$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}}}{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}} \cdot \frac{{(4 x + 1)}^{\frac{1}{2}} (e^{x^{2}} (2 x + 1) (6 (3 x + 1)) + {(3 x + 1)}^{2} (2 e^{x^{2}} + (2 x + 1) e^{x^{2}} (2 x))) - \frac{{(3 x + 1)}^{2} e^{x^{2}}}{2 {(4 x + 1)}^{\frac{1}{2}}} (2 x + 1) \frac{d}{d x} [4 x + 1]}{4 x + 1}$

Step 4.20

By the Sum Rule, the derivative of $4 x + 1$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [1]$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}}}{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}} \cdot \frac{{(4 x + 1)}^{\frac{1}{2}} (e^{x^{2}} (2 x + 1) (6 (3 x + 1)) + {(3 x + 1)}^{2} (2 e^{x^{2}} + (2 x + 1) e^{x^{2}} (2 x))) - \frac{{(3 x + 1)}^{2} e^{x^{2}}}{2 {(4 x + 1)}^{\frac{1}{2}}} (2 x + 1) (\frac{d}{d x} [4 x] + \frac{d}{d x} [1])}{4 x + 1}$

Step 4.21

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}}}{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}} \cdot \frac{{(4 x + 1)}^{\frac{1}{2}} (e^{x^{2}} (2 x + 1) (6 (3 x + 1)) + {(3 x + 1)}^{2} (2 e^{x^{2}} + (2 x + 1) e^{x^{2}} (2 x))) - \frac{{(3 x + 1)}^{2} e^{x^{2}}}{2 {(4 x + 1)}^{\frac{1}{2}}} (2 x + 1) (4 \frac{d}{d x} [x] + \frac{d}{d x} [1])}{4 x + 1}$

Step 4.22

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}}}{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}} \cdot \frac{{(4 x + 1)}^{\frac{1}{2}} (e^{x^{2}} (2 x + 1) (6 (3 x + 1)) + {(3 x + 1)}^{2} (2 e^{x^{2}} + (2 x + 1) e^{x^{2}} (2 x))) - \frac{{(3 x + 1)}^{2} e^{x^{2}}}{2 {(4 x + 1)}^{\frac{1}{2}}} (2 x + 1) (4 \cdot 1 + \frac{d}{d x} [1])}{4 x + 1}$

Step 4.23

Multiply $4$ by $1$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}}}{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}} \cdot \frac{{(4 x + 1)}^{\frac{1}{2}} (e^{x^{2}} (2 x + 1) (6 (3 x + 1)) + {(3 x + 1)}^{2} (2 e^{x^{2}} + (2 x + 1) e^{x^{2}} (2 x))) - \frac{{(3 x + 1)}^{2} e^{x^{2}}}{2 {(4 x + 1)}^{\frac{1}{2}}} (2 x + 1) (4 + \frac{d}{d x} [1])}{4 x + 1}$

Step 4.24

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}}}{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}} \cdot \frac{{(4 x + 1)}^{\frac{1}{2}} (e^{x^{2}} (2 x + 1) (6 (3 x + 1)) + {(3 x + 1)}^{2} (2 e^{x^{2}} + (2 x + 1) e^{x^{2}} (2 x))) - \frac{{(3 x + 1)}^{2} e^{x^{2}}}{2 {(4 x + 1)}^{\frac{1}{2}}} (2 x + 1) (4 + 0)}{4 x + 1}$

Step 4.25

Simplify terms.

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Step 4.25.1

Add $4$ and $0$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}}}{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}} \cdot \frac{{(4 x + 1)}^{\frac{1}{2}} (e^{x^{2}} (2 x + 1) (6 (3 x + 1)) + {(3 x + 1)}^{2} (2 e^{x^{2}} + (2 x + 1) e^{x^{2}} (2 x))) - \frac{{(3 x + 1)}^{2} e^{x^{2}}}{2 {(4 x + 1)}^{\frac{1}{2}}} (2 x + 1) \cdot 4}{4 x + 1}$

Step 4.25.2

Multiply $4$ by $- 1$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}}}{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}} \cdot \frac{{(4 x + 1)}^{\frac{1}{2}} (e^{x^{2}} (2 x + 1) (6 (3 x + 1)) + {(3 x + 1)}^{2} (2 e^{x^{2}} + (2 x + 1) e^{x^{2}} (2 x))) - 4 \frac{{(3 x + 1)}^{2} e^{x^{2}}}{2 {(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{4 x + 1}$

Step 4.25.3

Combine $- 4$ and $\frac{{(3 x + 1)}^{2} e^{x^{2}}}{2 {(4 x + 1)}^{\frac{1}{2}}}$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}}}{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}} \cdot \frac{{(4 x + 1)}^{\frac{1}{2}} (e^{x^{2}} (2 x + 1) (6 (3 x + 1)) + {(3 x + 1)}^{2} (2 e^{x^{2}} + (2 x + 1) e^{x^{2}} (2 x))) + \frac{- 4 ({(3 x + 1)}^{2} e^{x^{2}})}{2 {(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{4 x + 1}$

Step 4.25.4

Factor $2$ out of $- 4 {(3 x + 1)}^{2} e^{x^{2}}$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}}}{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}} \cdot \frac{{(4 x + 1)}^{\frac{1}{2}} (e^{x^{2}} (2 x + 1) (6 (3 x + 1)) + {(3 x + 1)}^{2} (2 e^{x^{2}} + (2 x + 1) e^{x^{2}} (2 x))) + \frac{2 (- 2 {(3 x + 1)}^{2} e^{x^{2}})}{2 {(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{4 x + 1}$

Step 4.26

Cancel the common factors.

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Step 4.26.1

Factor $2$ out of $2 {(4 x + 1)}^{\frac{1}{2}}$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}}}{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}} \cdot \frac{{(4 x + 1)}^{\frac{1}{2}} (e^{x^{2}} (2 x + 1) (6 (3 x + 1)) + {(3 x + 1)}^{2} (2 e^{x^{2}} + (2 x + 1) e^{x^{2}} (2 x))) + \frac{2 (- 2 {(3 x + 1)}^{2} e^{x^{2}})}{2 ({(4 x + 1)}^{\frac{1}{2}})} (2 x + 1)}{4 x + 1}$

Step 4.26.2

Cancel the common factor.

$\frac{{(4 x + 1)}^{\frac{1}{2}}}{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}} \cdot \frac{{(4 x + 1)}^{\frac{1}{2}} (e^{x^{2}} (2 x + 1) (6 (3 x + 1)) + {(3 x + 1)}^{2} (2 e^{x^{2}} + (2 x + 1) e^{x^{2}} (2 x))) + \frac{2 (- 2 {(3 x + 1)}^{2} e^{x^{2}})}{2 {(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{4 x + 1}$

Step 4.26.3

Rewrite the expression.

$\frac{{(4 x + 1)}^{\frac{1}{2}}}{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}} \cdot \frac{{(4 x + 1)}^{\frac{1}{2}} (e^{x^{2}} (2 x + 1) (6 (3 x + 1)) + {(3 x + 1)}^{2} (2 e^{x^{2}} + (2 x + 1) e^{x^{2}} (2 x))) + \frac{- 2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{4 x + 1}$

Step 4.27

Move the negative in front of the fraction.

$\frac{{(4 x + 1)}^{\frac{1}{2}}}{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}} \cdot \frac{{(4 x + 1)}^{\frac{1}{2}} (e^{x^{2}} (2 x + 1) (6 (3 x + 1)) + {(3 x + 1)}^{2} (2 e^{x^{2}} + (2 x + 1) e^{x^{2}} (2 x))) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{4 x + 1}$

Step 4.28

Multiply $\frac{{(4 x + 1)}^{\frac{1}{2}}}{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2}}$ by $\frac{{(4 x + 1)}^{\frac{1}{2}} (e^{x^{2}} (2 x + 1) (6 (3 x + 1)) + {(3 x + 1)}^{2} (2 e^{x^{2}} + (2 x + 1) e^{x^{2}} (2 x))) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{4 x + 1}$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}} ({(4 x + 1)}^{\frac{1}{2}} (e^{x^{2}} (2 x + 1) (6 (3 x + 1)) + {(3 x + 1)}^{2} (2 e^{x^{2}} + (2 x + 1) e^{x^{2}} (2 x))) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1))}{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2} (4 x + 1)}$

Step 4.29

Move ${(4 x + 1)}^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}} (e^{x^{2}} (2 x + 1) (6 (3 x + 1)) + {(3 x + 1)}^{2} (2 e^{x^{2}} + (2 x + 1) e^{x^{2}} (2 x))) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2} (4 x + 1) {(4 x + 1)}^{- \frac{1}{2}}}$

Step 4.30

Multiply $4 x + 1$ by ${(4 x + 1)}^{- \frac{1}{2}}$ by adding the exponents.

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Step 4.30.1

Move ${(4 x + 1)}^{- \frac{1}{2}}$ .

Step 4.30.2

Multiply ${(4 x + 1)}^{- \frac{1}{2}}$ by $4 x + 1$ .

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Step 4.30.2.1

Raise $4 x + 1$ to the power of $1$ .

Step 4.30.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

Step 4.30.3

Write $1$ as a fraction with a common denominator.

Step 4.30.4

Combine the numerators over the common denominator.

Step 4.30.5

Add $- 1$ and $2$ .

Step 4.31

Simplify.

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Step 4.31.1

Apply the distributive property.

$\frac{{(4 x + 1)}^{\frac{1}{2}} ((e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) (6 (3 x + 1)) + {(3 x + 1)}^{2} (2 e^{x^{2}} + (2 x + 1) e^{x^{2}} (2 x))) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.2

Apply the distributive property.

$\frac{{(4 x + 1)}^{\frac{1}{2}} ((e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) (6 (3 x) + 6 \cdot 1) + {(3 x + 1)}^{2} (2 e^{x^{2}} + (2 x + 1) e^{x^{2}} (2 x))) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.3

Apply the distributive property.

$\frac{{(4 x + 1)}^{\frac{1}{2}} ((e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) (6 (3 x) + 6 \cdot 1) + {(3 x + 1)}^{2} (2 e^{x^{2}} + (2 x e^{x^{2}} + 1 e^{x^{2}}) (2 x))) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.4

Apply the distributive property.

$\frac{{(4 x + 1)}^{\frac{1}{2}} ((e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) (6 (3 x) + 6 \cdot 1) + {(3 x + 1)}^{2} (2 e^{x^{2}} + 2 x e^{x^{2}} (2 x) + 1 e^{x^{2}} (2 x))) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{e^{x^{2}} (2 x + 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.5

Apply the distributive property.

$\frac{{(4 x + 1)}^{\frac{1}{2}} ((e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) (6 (3 x) + 6 \cdot 1) + {(3 x + 1)}^{2} (2 e^{x^{2}} + 2 x e^{x^{2}} (2 x) + 1 e^{x^{2}} (2 x))) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6

Simplify the numerator.

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Step 4.31.6.1

Simplify each term.

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Step 4.31.6.1.1

Simplify each term.

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Step 4.31.6.1.1.1

Rewrite using the commutative property of multiplication.

$\frac{{(4 x + 1)}^{\frac{1}{2}} ((2 e^{x^{2}} x + e^{x^{2}} \cdot 1) (6 (3 x) + 6 \cdot 1) + {(3 x + 1)}^{2} (2 e^{x^{2}} + 2 x e^{x^{2}} (2 x) + 1 e^{x^{2}} (2 x))) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.1.2

Multiply $e^{x^{2}}$ by $1$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}} ((2 e^{x^{2}} x + e^{x^{2}}) (6 (3 x) + 6 \cdot 1) + {(3 x + 1)}^{2} (2 e^{x^{2}} + 2 x e^{x^{2}} (2 x) + 1 e^{x^{2}} (2 x))) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.2

Simplify each term.

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Step 4.31.6.1.2.1

Multiply $3$ by $6$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}} ((2 e^{x^{2}} x + e^{x^{2}}) (18 x + 6 \cdot 1) + {(3 x + 1)}^{2} (2 e^{x^{2}} + 2 x e^{x^{2}} (2 x) + 1 e^{x^{2}} (2 x))) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.2.2

Multiply $6$ by $1$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}} ((2 e^{x^{2}} x + e^{x^{2}}) (18 x + 6) + {(3 x + 1)}^{2} (2 e^{x^{2}} + 2 x e^{x^{2}} (2 x) + 1 e^{x^{2}} (2 x))) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.3

Expand $(2 e^{x^{2}} x + e^{x^{2}}) (18 x + 6)$ using the FOIL Method.

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Step 4.31.6.1.3.1

Apply the distributive property.

$\frac{{(4 x + 1)}^{\frac{1}{2}} (2 e^{x^{2}} x (18 x + 6) + e^{x^{2}} (18 x + 6) + {(3 x + 1)}^{2} (2 e^{x^{2}} + 2 x e^{x^{2}} (2 x) + 1 e^{x^{2}} (2 x))) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.3.2

Apply the distributive property.

$\frac{{(4 x + 1)}^{\frac{1}{2}} (2 e^{x^{2}} x (18 x) + 2 e^{x^{2}} x \cdot 6 + e^{x^{2}} (18 x + 6) + {(3 x + 1)}^{2} (2 e^{x^{2}} + 2 x e^{x^{2}} (2 x) + 1 e^{x^{2}} (2 x))) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.3.3

Apply the distributive property.

$\frac{{(4 x + 1)}^{\frac{1}{2}} (2 e^{x^{2}} x (18 x) + 2 e^{x^{2}} x \cdot 6 + e^{x^{2}} (18 x) + e^{x^{2}} \cdot 6 + {(3 x + 1)}^{2} (2 e^{x^{2}} + 2 x e^{x^{2}} (2 x) + 1 e^{x^{2}} (2 x))) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.4

Simplify and combine like terms.

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Step 4.31.6.1.4.1

Simplify each term.

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Step 4.31.6.1.4.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 4.31.6.1.4.1.1.1

Move $x$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}} (2 e^{x^{2}} (x \cdot x) \cdot 18 + 2 e^{x^{2}} x \cdot 6 + e^{x^{2}} (18 x) + e^{x^{2}} \cdot 6 + {(3 x + 1)}^{2} (2 e^{x^{2}} + 2 x e^{x^{2}} (2 x) + 1 e^{x^{2}} (2 x))) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.4.1.1.2

Multiply $x$ by $x$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}} (2 e^{x^{2}} x^{2} \cdot 18 + 2 e^{x^{2}} x \cdot 6 + e^{x^{2}} (18 x) + e^{x^{2}} \cdot 6 + {(3 x + 1)}^{2} (2 e^{x^{2}} + 2 x e^{x^{2}} (2 x) + 1 e^{x^{2}} (2 x))) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.4.1.2

Multiply $18$ by $2$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}} (36 e^{x^{2}} x^{2} + 2 e^{x^{2}} x \cdot 6 + e^{x^{2}} (18 x) + e^{x^{2}} \cdot 6 + {(3 x + 1)}^{2} (2 e^{x^{2}} + 2 x e^{x^{2}} (2 x) + 1 e^{x^{2}} (2 x))) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.4.1.3

Multiply $6$ by $2$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}} (36 e^{x^{2}} x^{2} + 12 e^{x^{2}} x + e^{x^{2}} (18 x) + e^{x^{2}} \cdot 6 + {(3 x + 1)}^{2} (2 e^{x^{2}} + 2 x e^{x^{2}} (2 x) + 1 e^{x^{2}} (2 x))) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.4.1.4

Rewrite using the commutative property of multiplication.

$\frac{{(4 x + 1)}^{\frac{1}{2}} (36 e^{x^{2}} x^{2} + 12 e^{x^{2}} x + 18 e^{x^{2}} x + e^{x^{2}} \cdot 6 + {(3 x + 1)}^{2} (2 e^{x^{2}} + 2 x e^{x^{2}} (2 x) + 1 e^{x^{2}} (2 x))) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.4.1.5

Move $6$ to the left of $e^{x^{2}}$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}} (36 e^{x^{2}} x^{2} + 12 e^{x^{2}} x + 18 e^{x^{2}} x + 6 e^{x^{2}} + {(3 x + 1)}^{2} (2 e^{x^{2}} + 2 x e^{x^{2}} (2 x) + 1 e^{x^{2}} (2 x))) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.4.2

Add $12 e^{x^{2}} x$ and $18 e^{x^{2}} x$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}} (36 e^{x^{2}} x^{2} + 30 e^{x^{2}} x + 6 e^{x^{2}} + {(3 x + 1)}^{2} (2 e^{x^{2}} + 2 x e^{x^{2}} (2 x) + 1 e^{x^{2}} (2 x))) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.5

Rewrite ${(3 x + 1)}^{2}$ as $(3 x + 1) (3 x + 1)$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}} (36 e^{x^{2}} x^{2} + 30 e^{x^{2}} x + 6 e^{x^{2}} + (3 x + 1) (3 x + 1) (2 e^{x^{2}} + 2 x e^{x^{2}} (2 x) + 1 e^{x^{2}} (2 x))) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.6

Expand $(3 x + 1) (3 x + 1)$ using the FOIL Method.

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Step 4.31.6.1.6.1

Apply the distributive property.

$\frac{{(4 x + 1)}^{\frac{1}{2}} (36 e^{x^{2}} x^{2} + 30 e^{x^{2}} x + 6 e^{x^{2}} + (3 x (3 x + 1) + 1 (3 x + 1)) (2 e^{x^{2}} + 2 x e^{x^{2}} (2 x) + 1 e^{x^{2}} (2 x))) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.6.2

Apply the distributive property.

$\frac{{(4 x + 1)}^{\frac{1}{2}} (36 e^{x^{2}} x^{2} + 30 e^{x^{2}} x + 6 e^{x^{2}} + (3 x (3 x) + 3 x \cdot 1 + 1 (3 x + 1)) (2 e^{x^{2}} + 2 x e^{x^{2}} (2 x) + 1 e^{x^{2}} (2 x))) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.6.3

Apply the distributive property.

$\frac{{(4 x + 1)}^{\frac{1}{2}} (36 e^{x^{2}} x^{2} + 30 e^{x^{2}} x + 6 e^{x^{2}} + (3 x (3 x) + 3 x \cdot 1 + 1 (3 x) + 1 \cdot 1) (2 e^{x^{2}} + 2 x e^{x^{2}} (2 x) + 1 e^{x^{2}} (2 x))) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.7

Simplify and combine like terms.

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Step 4.31.6.1.7.1

Simplify each term.

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Step 4.31.6.1.7.1.1

Rewrite using the commutative property of multiplication.

$\frac{{(4 x + 1)}^{\frac{1}{2}} (36 e^{x^{2}} x^{2} + 30 e^{x^{2}} x + 6 e^{x^{2}} + (3 \cdot 3 x \cdot x + 3 x \cdot 1 + 1 (3 x) + 1 \cdot 1) (2 e^{x^{2}} + 2 x e^{x^{2}} (2 x) + 1 e^{x^{2}} (2 x))) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.7.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 4.31.6.1.7.1.2.1

Move $x$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}} (36 e^{x^{2}} x^{2} + 30 e^{x^{2}} x + 6 e^{x^{2}} + (3 \cdot 3 (x \cdot x) + 3 x \cdot 1 + 1 (3 x) + 1 \cdot 1) (2 e^{x^{2}} + 2 x e^{x^{2}} (2 x) + 1 e^{x^{2}} (2 x))) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.7.1.2.2

Multiply $x$ by $x$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}} (36 e^{x^{2}} x^{2} + 30 e^{x^{2}} x + 6 e^{x^{2}} + (3 \cdot 3 x^{2} + 3 x \cdot 1 + 1 (3 x) + 1 \cdot 1) (2 e^{x^{2}} + 2 x e^{x^{2}} (2 x) + 1 e^{x^{2}} (2 x))) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.7.1.3

Multiply $3$ by $3$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}} (36 e^{x^{2}} x^{2} + 30 e^{x^{2}} x + 6 e^{x^{2}} + (9 x^{2} + 3 x \cdot 1 + 1 (3 x) + 1 \cdot 1) (2 e^{x^{2}} + 2 x e^{x^{2}} (2 x) + 1 e^{x^{2}} (2 x))) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.7.1.4

Multiply $3$ by $1$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}} (36 e^{x^{2}} x^{2} + 30 e^{x^{2}} x + 6 e^{x^{2}} + (9 x^{2} + 3 x + 1 (3 x) + 1 \cdot 1) (2 e^{x^{2}} + 2 x e^{x^{2}} (2 x) + 1 e^{x^{2}} (2 x))) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.7.1.5

Multiply $3 x$ by $1$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}} (36 e^{x^{2}} x^{2} + 30 e^{x^{2}} x + 6 e^{x^{2}} + (9 x^{2} + 3 x + 3 x + 1 \cdot 1) (2 e^{x^{2}} + 2 x e^{x^{2}} (2 x) + 1 e^{x^{2}} (2 x))) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.7.1.6

Multiply $1$ by $1$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}} (36 e^{x^{2}} x^{2} + 30 e^{x^{2}} x + 6 e^{x^{2}} + (9 x^{2} + 3 x + 3 x + 1) (2 e^{x^{2}} + 2 x e^{x^{2}} (2 x) + 1 e^{x^{2}} (2 x))) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.7.2

Add $3 x$ and $3 x$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}} (36 e^{x^{2}} x^{2} + 30 e^{x^{2}} x + 6 e^{x^{2}} + (9 x^{2} + 6 x + 1) (2 e^{x^{2}} + 2 x e^{x^{2}} (2 x) + 1 e^{x^{2}} (2 x))) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.8

Simplify each term.

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Step 4.31.6.1.8.1

Multiply $x$ by $x$ by adding the exponents.

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Step 4.31.6.1.8.1.1

Move $x$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}} (36 e^{x^{2}} x^{2} + 30 e^{x^{2}} x + 6 e^{x^{2}} + (9 x^{2} + 6 x + 1) (2 e^{x^{2}} + 2 (x \cdot x) e^{x^{2}} \cdot 2 + 1 e^{x^{2}} (2 x))) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.8.1.2

Multiply $x$ by $x$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}} (36 e^{x^{2}} x^{2} + 30 e^{x^{2}} x + 6 e^{x^{2}} + (9 x^{2} + 6 x + 1) (2 e^{x^{2}} + 2 x^{2} e^{x^{2}} \cdot 2 + 1 e^{x^{2}} (2 x))) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.8.2

Multiply $2$ by $2$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}} (36 e^{x^{2}} x^{2} + 30 e^{x^{2}} x + 6 e^{x^{2}} + (9 x^{2} + 6 x + 1) (2 e^{x^{2}} + 4 x^{2} e^{x^{2}} + 1 e^{x^{2}} (2 x))) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.8.3

Rewrite using the commutative property of multiplication.

$\frac{{(4 x + 1)}^{\frac{1}{2}} (36 e^{x^{2}} x^{2} + 30 e^{x^{2}} x + 6 e^{x^{2}} + (9 x^{2} + 6 x + 1) (2 e^{x^{2}} + 4 x^{2} e^{x^{2}} + 1 \cdot 2 e^{x^{2}} x)) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.8.4

Multiply $2$ by $1$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}} (36 e^{x^{2}} x^{2} + 30 e^{x^{2}} x + 6 e^{x^{2}} + (9 x^{2} + 6 x + 1) (2 e^{x^{2}} + 4 x^{2} e^{x^{2}} + 2 e^{x^{2}} x)) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.9

Expand $(9 x^{2} + 6 x + 1) (2 e^{x^{2}} + 4 x^{2} e^{x^{2}} + 2 e^{x^{2}} x)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{{(4 x + 1)}^{\frac{1}{2}} (36 e^{x^{2}} x^{2} + 30 e^{x^{2}} x + 6 e^{x^{2}} + 9 x^{2} (2 e^{x^{2}}) + 9 x^{2} (4 x^{2} e^{x^{2}}) + 9 x^{2} (2 e^{x^{2}} x) + 6 x (2 e^{x^{2}}) + 6 x (4 x^{2} e^{x^{2}}) + 6 x (2 e^{x^{2}} x) + 1 (2 e^{x^{2}}) + 1 (4 x^{2} e^{x^{2}}) + 1 (2 e^{x^{2}} x)) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.10

Simplify each term.

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Step 4.31.6.1.10.1

Rewrite using the commutative property of multiplication.

$\frac{{(4 x + 1)}^{\frac{1}{2}} (36 e^{x^{2}} x^{2} + 30 e^{x^{2}} x + 6 e^{x^{2}} + 9 \cdot 2 x^{2} e^{x^{2}} + 9 x^{2} (4 x^{2} e^{x^{2}}) + 9 x^{2} (2 e^{x^{2}} x) + 6 x (2 e^{x^{2}}) + 6 x (4 x^{2} e^{x^{2}}) + 6 x (2 e^{x^{2}} x) + 1 (2 e^{x^{2}}) + 1 (4 x^{2} e^{x^{2}}) + 1 (2 e^{x^{2}} x)) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.10.2

Multiply $9$ by $2$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}} (36 e^{x^{2}} x^{2} + 30 e^{x^{2}} x + 6 e^{x^{2}} + 18 x^{2} e^{x^{2}} + 9 x^{2} (4 x^{2} e^{x^{2}}) + 9 x^{2} (2 e^{x^{2}} x) + 6 x (2 e^{x^{2}}) + 6 x (4 x^{2} e^{x^{2}}) + 6 x (2 e^{x^{2}} x) + 1 (2 e^{x^{2}}) + 1 (4 x^{2} e^{x^{2}}) + 1 (2 e^{x^{2}} x)) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.10.3

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 4.31.6.1.10.3.1

Move $x^{2}$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}} (36 e^{x^{2}} x^{2} + 30 e^{x^{2}} x + 6 e^{x^{2}} + 18 x^{2} e^{x^{2}} + 9 (x^{2} x^{2}) (4 e^{x^{2}}) + 9 x^{2} (2 e^{x^{2}} x) + 6 x (2 e^{x^{2}}) + 6 x (4 x^{2} e^{x^{2}}) + 6 x (2 e^{x^{2}} x) + 1 (2 e^{x^{2}}) + 1 (4 x^{2} e^{x^{2}}) + 1 (2 e^{x^{2}} x)) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.10.3.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{{(4 x + 1)}^{\frac{1}{2}} (36 e^{x^{2}} x^{2} + 30 e^{x^{2}} x + 6 e^{x^{2}} + 18 x^{2} e^{x^{2}} + 9 x^{2 + 2} (4 e^{x^{2}}) + 9 x^{2} (2 e^{x^{2}} x) + 6 x (2 e^{x^{2}}) + 6 x (4 x^{2} e^{x^{2}}) + 6 x (2 e^{x^{2}} x) + 1 (2 e^{x^{2}}) + 1 (4 x^{2} e^{x^{2}}) + 1 (2 e^{x^{2}} x)) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.10.3.3

Add $2$ and $2$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}} (36 e^{x^{2}} x^{2} + 30 e^{x^{2}} x + 6 e^{x^{2}} + 18 x^{2} e^{x^{2}} + 9 x^{4} (4 e^{x^{2}}) + 9 x^{2} (2 e^{x^{2}} x) + 6 x (2 e^{x^{2}}) + 6 x (4 x^{2} e^{x^{2}}) + 6 x (2 e^{x^{2}} x) + 1 (2 e^{x^{2}}) + 1 (4 x^{2} e^{x^{2}}) + 1 (2 e^{x^{2}} x)) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.10.4

Rewrite using the commutative property of multiplication.

$\frac{{(4 x + 1)}^{\frac{1}{2}} (36 e^{x^{2}} x^{2} + 30 e^{x^{2}} x + 6 e^{x^{2}} + 18 x^{2} e^{x^{2}} + 9 \cdot 4 x^{4} e^{x^{2}} + 9 x^{2} (2 e^{x^{2}} x) + 6 x (2 e^{x^{2}}) + 6 x (4 x^{2} e^{x^{2}}) + 6 x (2 e^{x^{2}} x) + 1 (2 e^{x^{2}}) + 1 (4 x^{2} e^{x^{2}}) + 1 (2 e^{x^{2}} x)) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.10.5

Multiply $9$ by $4$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}} (36 e^{x^{2}} x^{2} + 30 e^{x^{2}} x + 6 e^{x^{2}} + 18 x^{2} e^{x^{2}} + 36 x^{4} e^{x^{2}} + 9 x^{2} (2 e^{x^{2}} x) + 6 x (2 e^{x^{2}}) + 6 x (4 x^{2} e^{x^{2}}) + 6 x (2 e^{x^{2}} x) + 1 (2 e^{x^{2}}) + 1 (4 x^{2} e^{x^{2}}) + 1 (2 e^{x^{2}} x)) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.10.6

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 4.31.6.1.10.6.1

Move $x$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}} (36 e^{x^{2}} x^{2} + 30 e^{x^{2}} x + 6 e^{x^{2}} + 18 x^{2} e^{x^{2}} + 36 x^{4} e^{x^{2}} + 9 (x \cdot x^{2}) (2 e^{x^{2}}) + 6 x (2 e^{x^{2}}) + 6 x (4 x^{2} e^{x^{2}}) + 6 x (2 e^{x^{2}} x) + 1 (2 e^{x^{2}}) + 1 (4 x^{2} e^{x^{2}}) + 1 (2 e^{x^{2}} x)) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.10.6.2

Multiply $x$ by $x^{2}$ .

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Step 4.31.6.1.10.6.2.1

Raise $x$ to the power of $1$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}} (36 e^{x^{2}} x^{2} + 30 e^{x^{2}} x + 6 e^{x^{2}} + 18 x^{2} e^{x^{2}} + 36 x^{4} e^{x^{2}} + 9 (x^{1} x^{2}) (2 e^{x^{2}}) + 6 x (2 e^{x^{2}}) + 6 x (4 x^{2} e^{x^{2}}) + 6 x (2 e^{x^{2}} x) + 1 (2 e^{x^{2}}) + 1 (4 x^{2} e^{x^{2}}) + 1 (2 e^{x^{2}} x)) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.10.6.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{{(4 x + 1)}^{\frac{1}{2}} (36 e^{x^{2}} x^{2} + 30 e^{x^{2}} x + 6 e^{x^{2}} + 18 x^{2} e^{x^{2}} + 36 x^{4} e^{x^{2}} + 9 x^{1 + 2} (2 e^{x^{2}}) + 6 x (2 e^{x^{2}}) + 6 x (4 x^{2} e^{x^{2}}) + 6 x (2 e^{x^{2}} x) + 1 (2 e^{x^{2}}) + 1 (4 x^{2} e^{x^{2}}) + 1 (2 e^{x^{2}} x)) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.10.6.3

Add $1$ and $2$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}} (36 e^{x^{2}} x^{2} + 30 e^{x^{2}} x + 6 e^{x^{2}} + 18 x^{2} e^{x^{2}} + 36 x^{4} e^{x^{2}} + 9 x^{3} (2 e^{x^{2}}) + 6 x (2 e^{x^{2}}) + 6 x (4 x^{2} e^{x^{2}}) + 6 x (2 e^{x^{2}} x) + 1 (2 e^{x^{2}}) + 1 (4 x^{2} e^{x^{2}}) + 1 (2 e^{x^{2}} x)) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.10.7

Rewrite using the commutative property of multiplication.

$\frac{{(4 x + 1)}^{\frac{1}{2}} (36 e^{x^{2}} x^{2} + 30 e^{x^{2}} x + 6 e^{x^{2}} + 18 x^{2} e^{x^{2}} + 36 x^{4} e^{x^{2}} + 9 \cdot 2 x^{3} e^{x^{2}} + 6 x (2 e^{x^{2}}) + 6 x (4 x^{2} e^{x^{2}}) + 6 x (2 e^{x^{2}} x) + 1 (2 e^{x^{2}}) + 1 (4 x^{2} e^{x^{2}}) + 1 (2 e^{x^{2}} x)) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.10.8

Multiply $9$ by $2$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}} (36 e^{x^{2}} x^{2} + 30 e^{x^{2}} x + 6 e^{x^{2}} + 18 x^{2} e^{x^{2}} + 36 x^{4} e^{x^{2}} + 18 x^{3} e^{x^{2}} + 6 x (2 e^{x^{2}}) + 6 x (4 x^{2} e^{x^{2}}) + 6 x (2 e^{x^{2}} x) + 1 (2 e^{x^{2}}) + 1 (4 x^{2} e^{x^{2}}) + 1 (2 e^{x^{2}} x)) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.10.9

Rewrite using the commutative property of multiplication.

$\frac{{(4 x + 1)}^{\frac{1}{2}} (36 e^{x^{2}} x^{2} + 30 e^{x^{2}} x + 6 e^{x^{2}} + 18 x^{2} e^{x^{2}} + 36 x^{4} e^{x^{2}} + 18 x^{3} e^{x^{2}} + 6 \cdot 2 x e^{x^{2}} + 6 x (4 x^{2} e^{x^{2}}) + 6 x (2 e^{x^{2}} x) + 1 (2 e^{x^{2}}) + 1 (4 x^{2} e^{x^{2}}) + 1 (2 e^{x^{2}} x)) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.10.10

Multiply $6$ by $2$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}} (36 e^{x^{2}} x^{2} + 30 e^{x^{2}} x + 6 e^{x^{2}} + 18 x^{2} e^{x^{2}} + 36 x^{4} e^{x^{2}} + 18 x^{3} e^{x^{2}} + 12 x e^{x^{2}} + 6 x (4 x^{2} e^{x^{2}}) + 6 x (2 e^{x^{2}} x) + 1 (2 e^{x^{2}}) + 1 (4 x^{2} e^{x^{2}}) + 1 (2 e^{x^{2}} x)) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.10.11

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 4.31.6.1.10.11.1

Move $x^{2}$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}} (36 e^{x^{2}} x^{2} + 30 e^{x^{2}} x + 6 e^{x^{2}} + 18 x^{2} e^{x^{2}} + 36 x^{4} e^{x^{2}} + 18 x^{3} e^{x^{2}} + 12 x e^{x^{2}} + 6 (x^{2} x) (4 e^{x^{2}}) + 6 x (2 e^{x^{2}} x) + 1 (2 e^{x^{2}}) + 1 (4 x^{2} e^{x^{2}}) + 1 (2 e^{x^{2}} x)) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.10.11.2

Multiply $x^{2}$ by $x$ .

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Step 4.31.6.1.10.11.2.1

Raise $x$ to the power of $1$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}} (36 e^{x^{2}} x^{2} + 30 e^{x^{2}} x + 6 e^{x^{2}} + 18 x^{2} e^{x^{2}} + 36 x^{4} e^{x^{2}} + 18 x^{3} e^{x^{2}} + 12 x e^{x^{2}} + 6 (x^{2} x^{1}) (4 e^{x^{2}}) + 6 x (2 e^{x^{2}} x) + 1 (2 e^{x^{2}}) + 1 (4 x^{2} e^{x^{2}}) + 1 (2 e^{x^{2}} x)) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.10.11.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{{(4 x + 1)}^{\frac{1}{2}} (36 e^{x^{2}} x^{2} + 30 e^{x^{2}} x + 6 e^{x^{2}} + 18 x^{2} e^{x^{2}} + 36 x^{4} e^{x^{2}} + 18 x^{3} e^{x^{2}} + 12 x e^{x^{2}} + 6 x^{2 + 1} (4 e^{x^{2}}) + 6 x (2 e^{x^{2}} x) + 1 (2 e^{x^{2}}) + 1 (4 x^{2} e^{x^{2}}) + 1 (2 e^{x^{2}} x)) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.10.11.3

Add $2$ and $1$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}} (36 e^{x^{2}} x^{2} + 30 e^{x^{2}} x + 6 e^{x^{2}} + 18 x^{2} e^{x^{2}} + 36 x^{4} e^{x^{2}} + 18 x^{3} e^{x^{2}} + 12 x e^{x^{2}} + 6 x^{3} (4 e^{x^{2}}) + 6 x (2 e^{x^{2}} x) + 1 (2 e^{x^{2}}) + 1 (4 x^{2} e^{x^{2}}) + 1 (2 e^{x^{2}} x)) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.10.12

Rewrite using the commutative property of multiplication.

$\frac{{(4 x + 1)}^{\frac{1}{2}} (36 e^{x^{2}} x^{2} + 30 e^{x^{2}} x + 6 e^{x^{2}} + 18 x^{2} e^{x^{2}} + 36 x^{4} e^{x^{2}} + 18 x^{3} e^{x^{2}} + 12 x e^{x^{2}} + 6 \cdot 4 x^{3} e^{x^{2}} + 6 x (2 e^{x^{2}} x) + 1 (2 e^{x^{2}}) + 1 (4 x^{2} e^{x^{2}}) + 1 (2 e^{x^{2}} x)) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.10.13

Multiply $6$ by $4$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}} (36 e^{x^{2}} x^{2} + 30 e^{x^{2}} x + 6 e^{x^{2}} + 18 x^{2} e^{x^{2}} + 36 x^{4} e^{x^{2}} + 18 x^{3} e^{x^{2}} + 12 x e^{x^{2}} + 24 x^{3} e^{x^{2}} + 6 x (2 e^{x^{2}} x) + 1 (2 e^{x^{2}}) + 1 (4 x^{2} e^{x^{2}}) + 1 (2 e^{x^{2}} x)) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.10.14

Multiply $x$ by $x$ by adding the exponents.

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Step 4.31.6.1.10.14.1

Move $x$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}} (36 e^{x^{2}} x^{2} + 30 e^{x^{2}} x + 6 e^{x^{2}} + 18 x^{2} e^{x^{2}} + 36 x^{4} e^{x^{2}} + 18 x^{3} e^{x^{2}} + 12 x e^{x^{2}} + 24 x^{3} e^{x^{2}} + 6 (x \cdot x) (2 e^{x^{2}}) + 1 (2 e^{x^{2}}) + 1 (4 x^{2} e^{x^{2}}) + 1 (2 e^{x^{2}} x)) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.10.14.2

Multiply $x$ by $x$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}} (36 e^{x^{2}} x^{2} + 30 e^{x^{2}} x + 6 e^{x^{2}} + 18 x^{2} e^{x^{2}} + 36 x^{4} e^{x^{2}} + 18 x^{3} e^{x^{2}} + 12 x e^{x^{2}} + 24 x^{3} e^{x^{2}} + 6 x^{2} (2 e^{x^{2}}) + 1 (2 e^{x^{2}}) + 1 (4 x^{2} e^{x^{2}}) + 1 (2 e^{x^{2}} x)) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.10.15

Rewrite using the commutative property of multiplication.

$\frac{{(4 x + 1)}^{\frac{1}{2}} (36 e^{x^{2}} x^{2} + 30 e^{x^{2}} x + 6 e^{x^{2}} + 18 x^{2} e^{x^{2}} + 36 x^{4} e^{x^{2}} + 18 x^{3} e^{x^{2}} + 12 x e^{x^{2}} + 24 x^{3} e^{x^{2}} + 6 \cdot 2 x^{2} e^{x^{2}} + 1 (2 e^{x^{2}}) + 1 (4 x^{2} e^{x^{2}}) + 1 (2 e^{x^{2}} x)) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.10.16

Multiply $6$ by $2$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}} (36 e^{x^{2}} x^{2} + 30 e^{x^{2}} x + 6 e^{x^{2}} + 18 x^{2} e^{x^{2}} + 36 x^{4} e^{x^{2}} + 18 x^{3} e^{x^{2}} + 12 x e^{x^{2}} + 24 x^{3} e^{x^{2}} + 12 x^{2} e^{x^{2}} + 1 (2 e^{x^{2}}) + 1 (4 x^{2} e^{x^{2}}) + 1 (2 e^{x^{2}} x)) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.10.17

Multiply $2 e^{x^{2}}$ by $1$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}} (36 e^{x^{2}} x^{2} + 30 e^{x^{2}} x + 6 e^{x^{2}} + 18 x^{2} e^{x^{2}} + 36 x^{4} e^{x^{2}} + 18 x^{3} e^{x^{2}} + 12 x e^{x^{2}} + 24 x^{3} e^{x^{2}} + 12 x^{2} e^{x^{2}} + 2 e^{x^{2}} + 1 (4 x^{2} e^{x^{2}}) + 1 (2 e^{x^{2}} x)) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.10.18

Multiply $4 x^{2} e^{x^{2}}$ by $1$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}} (36 e^{x^{2}} x^{2} + 30 e^{x^{2}} x + 6 e^{x^{2}} + 18 x^{2} e^{x^{2}} + 36 x^{4} e^{x^{2}} + 18 x^{3} e^{x^{2}} + 12 x e^{x^{2}} + 24 x^{3} e^{x^{2}} + 12 x^{2} e^{x^{2}} + 2 e^{x^{2}} + 4 x^{2} e^{x^{2}} + 1 (2 e^{x^{2}} x)) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.10.19

Multiply $2 e^{x^{2}} x$ by $1$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}} (36 e^{x^{2}} x^{2} + 30 e^{x^{2}} x + 6 e^{x^{2}} + 18 x^{2} e^{x^{2}} + 36 x^{4} e^{x^{2}} + 18 x^{3} e^{x^{2}} + 12 x e^{x^{2}} + 24 x^{3} e^{x^{2}} + 12 x^{2} e^{x^{2}} + 2 e^{x^{2}} + 4 x^{2} e^{x^{2}} + 2 e^{x^{2}} x) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.11

Add $18 x^{2} e^{x^{2}}$ and $12 x^{2} e^{x^{2}}$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}} (36 e^{x^{2}} x^{2} + 30 e^{x^{2}} x + 6 e^{x^{2}} + 30 x^{2} e^{x^{2}} + 36 x^{4} e^{x^{2}} + 18 x^{3} e^{x^{2}} + 12 x e^{x^{2}} + 24 x^{3} e^{x^{2}} + 2 e^{x^{2}} + 4 x^{2} e^{x^{2}} + 2 e^{x^{2}} x) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.12

Add $30 x^{2} e^{x^{2}}$ and $4 x^{2} e^{x^{2}}$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}} (36 e^{x^{2}} x^{2} + 30 e^{x^{2}} x + 6 e^{x^{2}} + 34 x^{2} e^{x^{2}} + 36 x^{4} e^{x^{2}} + 18 x^{3} e^{x^{2}} + 12 x e^{x^{2}} + 24 x^{3} e^{x^{2}} + 2 e^{x^{2}} + 2 e^{x^{2}} x) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.13

Add $18 x^{3} e^{x^{2}}$ and $24 x^{3} e^{x^{2}}$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}} (36 e^{x^{2}} x^{2} + 30 e^{x^{2}} x + 6 e^{x^{2}} + 34 x^{2} e^{x^{2}} + 36 x^{4} e^{x^{2}} + 42 x^{3} e^{x^{2}} + 12 x e^{x^{2}} + 2 e^{x^{2}} + 2 e^{x^{2}} x) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.14

Add $12 x e^{x^{2}}$ and $2 e^{x^{2}} x$ .

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Step 4.31.6.1.14.1

Move $e^{x^{2}}$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}} (36 e^{x^{2}} x^{2} + 30 e^{x^{2}} x + 6 e^{x^{2}} + 34 x^{2} e^{x^{2}} + 36 x^{4} e^{x^{2}} + 42 x^{3} e^{x^{2}} + 12 x e^{x^{2}} + 2 x e^{x^{2}} + 2 e^{x^{2}}) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.1.14.2

Add $12 x e^{x^{2}}$ and $2 x e^{x^{2}}$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}} (36 e^{x^{2}} x^{2} + 30 e^{x^{2}} x + 6 e^{x^{2}} + 34 x^{2} e^{x^{2}} + 36 x^{4} e^{x^{2}} + 42 x^{3} e^{x^{2}} + 14 x e^{x^{2}} + 2 e^{x^{2}}) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.2

Add $36 e^{x^{2}} x^{2}$ and $34 x^{2} e^{x^{2}}$ .

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Step 4.31.6.2.1

Move $e^{x^{2}}$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}} (36 x^{2} e^{x^{2}} + 34 x^{2} e^{x^{2}} + 30 e^{x^{2}} x + 6 e^{x^{2}} + 36 x^{4} e^{x^{2}} + 42 x^{3} e^{x^{2}} + 14 x e^{x^{2}} + 2 e^{x^{2}}) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.2.2

Add $36 x^{2} e^{x^{2}}$ and $34 x^{2} e^{x^{2}}$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}} (70 x^{2} e^{x^{2}} + 30 e^{x^{2}} x + 6 e^{x^{2}} + 36 x^{4} e^{x^{2}} + 42 x^{3} e^{x^{2}} + 14 x e^{x^{2}} + 2 e^{x^{2}}) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.3

Add $30 e^{x^{2}} x$ and $14 x e^{x^{2}}$ .

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Step 4.31.6.3.1

Move $e^{x^{2}}$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}} (70 x^{2} e^{x^{2}} + 30 x e^{x^{2}} + 14 x e^{x^{2}} + 6 e^{x^{2}} + 36 x^{4} e^{x^{2}} + 42 x^{3} e^{x^{2}} + 2 e^{x^{2}}) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.3.2

Add $30 x e^{x^{2}}$ and $14 x e^{x^{2}}$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}} (70 x^{2} e^{x^{2}} + 44 x e^{x^{2}} + 6 e^{x^{2}} + 36 x^{4} e^{x^{2}} + 42 x^{3} e^{x^{2}} + 2 e^{x^{2}}) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.4

Add $6 e^{x^{2}}$ and $2 e^{x^{2}}$ .

$\frac{{(4 x + 1)}^{\frac{1}{2}} (70 x^{2} e^{x^{2}} + 44 x e^{x^{2}} + 36 x^{4} e^{x^{2}} + 42 x^{3} e^{x^{2}} + 8 e^{x^{2}}) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.5

Apply the distributive property.

$\frac{{(4 x + 1)}^{\frac{1}{2}} (70 x^{2} e^{x^{2}}) + {(4 x + 1)}^{\frac{1}{2}} (44 x e^{x^{2}}) + {(4 x + 1)}^{\frac{1}{2}} (36 x^{4} e^{x^{2}}) + {(4 x + 1)}^{\frac{1}{2}} (42 x^{3} e^{x^{2}}) + {(4 x + 1)}^{\frac{1}{2}} (8 e^{x^{2}}) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.6

Simplify.

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Step 4.31.6.6.1

Rewrite using the commutative property of multiplication.

$\frac{70 {(4 x + 1)}^{\frac{1}{2}} x^{2} e^{x^{2}} + {(4 x + 1)}^{\frac{1}{2}} (44 x e^{x^{2}}) + {(4 x + 1)}^{\frac{1}{2}} (36 x^{4} e^{x^{2}}) + {(4 x + 1)}^{\frac{1}{2}} (42 x^{3} e^{x^{2}}) + {(4 x + 1)}^{\frac{1}{2}} (8 e^{x^{2}}) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.6.2

Rewrite using the commutative property of multiplication.

$\frac{70 {(4 x + 1)}^{\frac{1}{2}} x^{2} e^{x^{2}} + 44 {(4 x + 1)}^{\frac{1}{2}} x e^{x^{2}} + {(4 x + 1)}^{\frac{1}{2}} (36 x^{4} e^{x^{2}}) + {(4 x + 1)}^{\frac{1}{2}} (42 x^{3} e^{x^{2}}) + {(4 x + 1)}^{\frac{1}{2}} (8 e^{x^{2}}) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.6.3

Rewrite using the commutative property of multiplication.

$\frac{70 {(4 x + 1)}^{\frac{1}{2}} x^{2} e^{x^{2}} + 44 {(4 x + 1)}^{\frac{1}{2}} x e^{x^{2}} + 36 {(4 x + 1)}^{\frac{1}{2}} x^{4} e^{x^{2}} + {(4 x + 1)}^{\frac{1}{2}} (42 x^{3} e^{x^{2}}) + {(4 x + 1)}^{\frac{1}{2}} (8 e^{x^{2}}) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.6.4

Rewrite using the commutative property of multiplication.

$\frac{70 {(4 x + 1)}^{\frac{1}{2}} x^{2} e^{x^{2}} + 44 {(4 x + 1)}^{\frac{1}{2}} x e^{x^{2}} + 36 {(4 x + 1)}^{\frac{1}{2}} x^{4} e^{x^{2}} + 42 {(4 x + 1)}^{\frac{1}{2}} x^{3} e^{x^{2}} + {(4 x + 1)}^{\frac{1}{2}} (8 e^{x^{2}}) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.6.5

Rewrite using the commutative property of multiplication.

$\frac{70 {(4 x + 1)}^{\frac{1}{2}} x^{2} e^{x^{2}} + 44 {(4 x + 1)}^{\frac{1}{2}} x e^{x^{2}} + 36 {(4 x + 1)}^{\frac{1}{2}} x^{4} e^{x^{2}} + 42 {(4 x + 1)}^{\frac{1}{2}} x^{3} e^{x^{2}} + 8 {(4 x + 1)}^{\frac{1}{2}} e^{x^{2}} - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x + 1)}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.7

Apply the distributive property.

$\frac{70 {(4 x + 1)}^{\frac{1}{2}} x^{2} e^{x^{2}} + 44 {(4 x + 1)}^{\frac{1}{2}} x e^{x^{2}} + 36 {(4 x + 1)}^{\frac{1}{2}} x^{4} e^{x^{2}} + 42 {(4 x + 1)}^{\frac{1}{2}} x^{3} e^{x^{2}} + 8 {(4 x + 1)}^{\frac{1}{2}} e^{x^{2}} - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x) - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} \cdot 1}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.8

Multiply $- \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} (2 x)$ .

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Step 4.31.6.8.1

Multiply $2$ by $- 1$ .

$\frac{70 {(4 x + 1)}^{\frac{1}{2}} x^{2} e^{x^{2}} + 44 {(4 x + 1)}^{\frac{1}{2}} x e^{x^{2}} + 36 {(4 x + 1)}^{\frac{1}{2}} x^{4} e^{x^{2}} + 42 {(4 x + 1)}^{\frac{1}{2}} x^{3} e^{x^{2}} + 8 {(4 x + 1)}^{\frac{1}{2}} e^{x^{2}} - 2 \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} x - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} \cdot 1}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.8.2

Combine $- 2$ and $\frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}}$ .

$\frac{70 {(4 x + 1)}^{\frac{1}{2}} x^{2} e^{x^{2}} + 44 {(4 x + 1)}^{\frac{1}{2}} x e^{x^{2}} + 36 {(4 x + 1)}^{\frac{1}{2}} x^{4} e^{x^{2}} + 42 {(4 x + 1)}^{\frac{1}{2}} x^{3} e^{x^{2}} + 8 {(4 x + 1)}^{\frac{1}{2}} e^{x^{2}} + \frac{- 2 (2 {(3 x + 1)}^{2} e^{x^{2}})}{{(4 x + 1)}^{\frac{1}{2}}} x - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} \cdot 1}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.8.3

Multiply $2$ by $- 2$ .

$\frac{70 {(4 x + 1)}^{\frac{1}{2}} x^{2} e^{x^{2}} + 44 {(4 x + 1)}^{\frac{1}{2}} x e^{x^{2}} + 36 {(4 x + 1)}^{\frac{1}{2}} x^{4} e^{x^{2}} + 42 {(4 x + 1)}^{\frac{1}{2}} x^{3} e^{x^{2}} + 8 {(4 x + 1)}^{\frac{1}{2}} e^{x^{2}} + \frac{- 4 ({(3 x + 1)}^{2} e^{x^{2}})}{{(4 x + 1)}^{\frac{1}{2}}} x - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} \cdot 1}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.8.4

Combine $\frac{- 4 ({(3 x + 1)}^{2} e^{x^{2}})}{{(4 x + 1)}^{\frac{1}{2}}}$ and $x$ .

$\frac{70 {(4 x + 1)}^{\frac{1}{2}} x^{2} e^{x^{2}} + 44 {(4 x + 1)}^{\frac{1}{2}} x e^{x^{2}} + 36 {(4 x + 1)}^{\frac{1}{2}} x^{4} e^{x^{2}} + 42 {(4 x + 1)}^{\frac{1}{2}} x^{3} e^{x^{2}} + 8 {(4 x + 1)}^{\frac{1}{2}} e^{x^{2}} + \frac{- 4 ({(3 x + 1)}^{2} e^{x^{2}}) x}{{(4 x + 1)}^{\frac{1}{2}}} - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}} \cdot 1}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.9

Multiply $- 1$ by $1$ .

$\frac{70 {(4 x + 1)}^{\frac{1}{2}} x^{2} e^{x^{2}} + 44 {(4 x + 1)}^{\frac{1}{2}} x e^{x^{2}} + 36 {(4 x + 1)}^{\frac{1}{2}} x^{4} e^{x^{2}} + 42 {(4 x + 1)}^{\frac{1}{2}} x^{3} e^{x^{2}} + 8 {(4 x + 1)}^{\frac{1}{2}} e^{x^{2}} + \frac{- 4 ({(3 x + 1)}^{2} e^{x^{2}}) x}{{(4 x + 1)}^{\frac{1}{2}}} - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.10

Move the negative in front of the fraction.

$\frac{70 {(4 x + 1)}^{\frac{1}{2}} x^{2} e^{x^{2}} + 44 {(4 x + 1)}^{\frac{1}{2}} x e^{x^{2}} + 36 {(4 x + 1)}^{\frac{1}{2}} x^{4} e^{x^{2}} + 42 {(4 x + 1)}^{\frac{1}{2}} x^{3} e^{x^{2}} + 8 {(4 x + 1)}^{\frac{1}{2}} e^{x^{2}} - \frac{4 {(3 x + 1)}^{2} e^{x^{2}} x}{{(4 x + 1)}^{\frac{1}{2}}} - \frac{2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.11

Combine the numerators over the common denominator.

$\frac{70 {(4 x + 1)}^{\frac{1}{2}} x^{2} e^{x^{2}} + 44 {(4 x + 1)}^{\frac{1}{2}} x e^{x^{2}} + 36 {(4 x + 1)}^{\frac{1}{2}} x^{4} e^{x^{2}} + 42 {(4 x + 1)}^{\frac{1}{2}} x^{3} e^{x^{2}} + 8 {(4 x + 1)}^{\frac{1}{2}} e^{x^{2}} + \frac{- 4 {(3 x + 1)}^{2} e^{x^{2}} x - 2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.12

Factor $2 {(3 x + 1)}^{2} e^{x^{2}}$ out of $- 4 {(3 x + 1)}^{2} e^{x^{2}} x - 2 {(3 x + 1)}^{2} e^{x^{2}}$ .

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Step 4.31.6.12.1

Factor $2 {(3 x + 1)}^{2} e^{x^{2}}$ out of $- 4 {(3 x + 1)}^{2} e^{x^{2}} x$ .

$\frac{70 {(4 x + 1)}^{\frac{1}{2}} x^{2} e^{x^{2}} + 44 {(4 x + 1)}^{\frac{1}{2}} x e^{x^{2}} + 36 {(4 x + 1)}^{\frac{1}{2}} x^{4} e^{x^{2}} + 42 {(4 x + 1)}^{\frac{1}{2}} x^{3} e^{x^{2}} + 8 {(4 x + 1)}^{\frac{1}{2}} e^{x^{2}} + \frac{2 {(3 x + 1)}^{2} e^{x^{2}} (- 2 x) - 2 {(3 x + 1)}^{2} e^{x^{2}}}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.12.2

Factor $2 {(3 x + 1)}^{2} e^{x^{2}}$ out of $- 2 {(3 x + 1)}^{2} e^{x^{2}}$ .

$\frac{70 {(4 x + 1)}^{\frac{1}{2}} x^{2} e^{x^{2}} + 44 {(4 x + 1)}^{\frac{1}{2}} x e^{x^{2}} + 36 {(4 x + 1)}^{\frac{1}{2}} x^{4} e^{x^{2}} + 42 {(4 x + 1)}^{\frac{1}{2}} x^{3} e^{x^{2}} + 8 {(4 x + 1)}^{\frac{1}{2}} e^{x^{2}} + \frac{2 {(3 x + 1)}^{2} e^{x^{2}} (- 2 x) + 2 {(3 x + 1)}^{2} e^{x^{2}} (- 1)}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.12.3

Factor $2 {(3 x + 1)}^{2} e^{x^{2}}$ out of $2 {(3 x + 1)}^{2} e^{x^{2}} (- 2 x) + 2 {(3 x + 1)}^{2} e^{x^{2}} (- 1)$ .

$\frac{70 {(4 x + 1)}^{\frac{1}{2}} x^{2} e^{x^{2}} + 44 {(4 x + 1)}^{\frac{1}{2}} x e^{x^{2}} + 36 {(4 x + 1)}^{\frac{1}{2}} x^{4} e^{x^{2}} + 42 {(4 x + 1)}^{\frac{1}{2}} x^{3} e^{x^{2}} + 8 {(4 x + 1)}^{\frac{1}{2}} e^{x^{2}} + \frac{2 {(3 x + 1)}^{2} e^{x^{2}} (- 2 x - 1)}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.13

To write $70 {(4 x + 1)}^{\frac{1}{2}} x^{2} e^{x^{2}}$ as a fraction with a common denominator, multiply by $\frac{{(4 x + 1)}^{\frac{1}{2}}}{{(4 x + 1)}^{\frac{1}{2}}}$ .

$\frac{44 {(4 x + 1)}^{\frac{1}{2}} x e^{x^{2}} + 36 {(4 x + 1)}^{\frac{1}{2}} x^{4} e^{x^{2}} + 42 {(4 x + 1)}^{\frac{1}{2}} x^{3} e^{x^{2}} + 8 {(4 x + 1)}^{\frac{1}{2}} e^{x^{2}} + \frac{70 {(4 x + 1)}^{\frac{1}{2}} x^{2} e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}}}{{(4 x + 1)}^{\frac{1}{2}}} + \frac{2 {(3 x + 1)}^{2} e^{x^{2}} (- 2 x - 1)}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.14

Combine the numerators over the common denominator.

$\frac{44 {(4 x + 1)}^{\frac{1}{2}} x e^{x^{2}} + 36 {(4 x + 1)}^{\frac{1}{2}} x^{4} e^{x^{2}} + 42 {(4 x + 1)}^{\frac{1}{2}} x^{3} e^{x^{2}} + 8 {(4 x + 1)}^{\frac{1}{2}} e^{x^{2}} + \frac{70 {(4 x + 1)}^{\frac{1}{2}} x^{2} e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}} + 2 {(3 x + 1)}^{2} e^{x^{2}} (- 2 x - 1)}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.15

Move ${(3 x + 1)}^{2}$ .

$\frac{36 {(4 x + 1)}^{\frac{1}{2}} x^{4} e^{x^{2}} + 42 {(4 x + 1)}^{\frac{1}{2}} x^{3} e^{x^{2}} + 8 {(4 x + 1)}^{\frac{1}{2}} e^{x^{2}} + 44 x e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}} + \frac{70 x^{2} e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 2 e^{x^{2}} {(3 x + 1)}^{2} (- 2 x - 1)}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.16

Add $36 {(4 x + 1)}^{\frac{1}{2}} x^{4} e^{x^{2}}$ and $\frac{70 x^{2} e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 2 e^{x^{2}} {(3 x + 1)}^{2} (- 2 x - 1)}{{(4 x + 1)}^{\frac{1}{2}}}$ .

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Step 4.31.6.16.1

Move ${(4 x + 1)}^{\frac{1}{2}}$ .

$\frac{42 {(4 x + 1)}^{\frac{1}{2}} x^{3} e^{x^{2}} + 8 {(4 x + 1)}^{\frac{1}{2}} e^{x^{2}} + 44 x e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}} + 36 x^{4} e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}} + \frac{70 x^{2} e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 2 e^{x^{2}} {(3 x + 1)}^{2} (- 2 x - 1)}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.16.2

To write $36 x^{4} e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{{(4 x + 1)}^{\frac{1}{2}}}{{(4 x + 1)}^{\frac{1}{2}}}$ .

$\frac{42 {(4 x + 1)}^{\frac{1}{2}} x^{3} e^{x^{2}} + 8 {(4 x + 1)}^{\frac{1}{2}} e^{x^{2}} + 44 x e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}} + \frac{36 x^{4} e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}}}{{(4 x + 1)}^{\frac{1}{2}}} + \frac{70 x^{2} e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 2 e^{x^{2}} {(3 x + 1)}^{2} (- 2 x - 1)}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.16.3

Combine the numerators over the common denominator.

$\frac{42 {(4 x + 1)}^{\frac{1}{2}} x^{3} e^{x^{2}} + 8 {(4 x + 1)}^{\frac{1}{2}} e^{x^{2}} + 44 x e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}} + \frac{36 x^{4} e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 70 x^{2} e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 2 e^{x^{2}} {(3 x + 1)}^{2} (- 2 x - 1)}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.17

Add $42 {(4 x + 1)}^{\frac{1}{2}} x^{3} e^{x^{2}}$ and $\frac{36 x^{4} e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 70 x^{2} e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 2 e^{x^{2}} {(3 x + 1)}^{2} (- 2 x - 1)}{{(4 x + 1)}^{\frac{1}{2}}}$ .

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Step 4.31.6.17.1

Move ${(4 x + 1)}^{\frac{1}{2}}$ .

$\frac{8 {(4 x + 1)}^{\frac{1}{2}} e^{x^{2}} + 44 x e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}} + 42 x^{3} e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}} + \frac{36 x^{4} e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 70 x^{2} e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 2 e^{x^{2}} {(3 x + 1)}^{2} (- 2 x - 1)}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.17.2

To write $42 x^{3} e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{{(4 x + 1)}^{\frac{1}{2}}}{{(4 x + 1)}^{\frac{1}{2}}}$ .

$\frac{8 {(4 x + 1)}^{\frac{1}{2}} e^{x^{2}} + 44 x e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}} + \frac{42 x^{3} e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}}}{{(4 x + 1)}^{\frac{1}{2}}} + \frac{36 x^{4} e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 70 x^{2} e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 2 e^{x^{2}} {(3 x + 1)}^{2} (- 2 x - 1)}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.17.3

Combine the numerators over the common denominator.

$\frac{8 {(4 x + 1)}^{\frac{1}{2}} e^{x^{2}} + 44 x e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}} + \frac{42 x^{3} e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 36 x^{4} e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 70 x^{2} e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 2 e^{x^{2}} {(3 x + 1)}^{2} (- 2 x - 1)}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.18

Add $8 {(4 x + 1)}^{\frac{1}{2}} e^{x^{2}}$ and $\frac{42 x^{3} e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 36 x^{4} e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 70 x^{2} e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 2 e^{x^{2}} {(3 x + 1)}^{2} (- 2 x - 1)}{{(4 x + 1)}^{\frac{1}{2}}}$ .

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Step 4.31.6.18.1

Move ${(4 x + 1)}^{\frac{1}{2}}$ .

$\frac{44 x e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}} + 8 e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}} + \frac{42 x^{3} e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 36 x^{4} e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 70 x^{2} e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 2 e^{x^{2}} {(3 x + 1)}^{2} (- 2 x - 1)}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.18.2

To write $8 e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{{(4 x + 1)}^{\frac{1}{2}}}{{(4 x + 1)}^{\frac{1}{2}}}$ .

$\frac{44 x e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}} + \frac{8 e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}}}{{(4 x + 1)}^{\frac{1}{2}}} + \frac{42 x^{3} e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 36 x^{4} e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 70 x^{2} e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 2 e^{x^{2}} {(3 x + 1)}^{2} (- 2 x - 1)}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.18.3

Combine the numerators over the common denominator.

$\frac{44 x e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}} + \frac{8 e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 42 x^{3} e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 36 x^{4} e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 70 x^{2} e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 2 e^{x^{2}} {(3 x + 1)}^{2} (- 2 x - 1)}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.19

To write $44 x e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{{(4 x + 1)}^{\frac{1}{2}}}{{(4 x + 1)}^{\frac{1}{2}}}$ .

$\frac{\frac{44 x e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}}}{{(4 x + 1)}^{\frac{1}{2}}} + \frac{8 e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 42 x^{3} e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 36 x^{4} e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 70 x^{2} e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 2 e^{x^{2}} {(3 x + 1)}^{2} (- 2 x - 1)}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.20

Combine the numerators over the common denominator.

$\frac{\frac{44 x e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 8 e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 42 x^{3} e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 36 x^{4} e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 70 x^{2} e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 2 e^{x^{2}} {(3 x + 1)}^{2} (- 2 x - 1)}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.21

Reorder terms.

$\frac{\frac{44 e^{x^{2}} x {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 8 e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 42 e^{x^{2}} x^{3} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 36 e^{x^{2}} x^{4} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 70 e^{x^{2}} x^{2} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 2 e^{x^{2}} {(3 x + 1)}^{2} (- 2 x - 1)}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.22

Rewrite $\frac{44 e^{x^{2}} x {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 8 e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 42 e^{x^{2}} x^{3} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 36 e^{x^{2}} x^{4} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 70 e^{x^{2}} x^{2} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 2 e^{x^{2}} {(3 x + 1)}^{2} (- 2 x - 1)}{{(4 x + 1)}^{\frac{1}{2}}}$ in a factored form.

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Step 4.31.6.22.1

Factor $2 e^{x^{2}}$ out of $44 e^{x^{2}} x {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 8 e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 42 e^{x^{2}} x^{3} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 36 e^{x^{2}} x^{4} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 70 e^{x^{2}} x^{2} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 2 e^{x^{2}} {(3 x + 1)}^{2} (- 2 x - 1)$ .

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Step 4.31.6.22.1.1

Factor $2 e^{x^{2}}$ out of $44 e^{x^{2}} x {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}}$ .

$\frac{\frac{2 e^{x^{2}} (22 x {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}}) + 8 e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 42 e^{x^{2}} x^{3} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 36 e^{x^{2}} x^{4} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 70 e^{x^{2}} x^{2} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 2 e^{x^{2}} {(3 x + 1)}^{2} (- 2 x - 1)}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.22.1.2

Factor $2 e^{x^{2}}$ out of $8 e^{x^{2}} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}}$ .

$\frac{\frac{2 e^{x^{2}} (22 x {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}}) + 2 e^{x^{2}} (4 {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}}) + 42 e^{x^{2}} x^{3} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 36 e^{x^{2}} x^{4} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 70 e^{x^{2}} x^{2} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 2 e^{x^{2}} {(3 x + 1)}^{2} (- 2 x - 1)}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.22.1.3

Factor $2 e^{x^{2}}$ out of $42 e^{x^{2}} x^{3} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}}$ .

$\frac{\frac{2 e^{x^{2}} (22 x {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}}) + 2 e^{x^{2}} (4 {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}}) + 2 e^{x^{2}} (21 x^{3} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}}) + 36 e^{x^{2}} x^{4} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 70 e^{x^{2}} x^{2} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 2 e^{x^{2}} {(3 x + 1)}^{2} (- 2 x - 1)}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.22.1.4

Factor $2 e^{x^{2}}$ out of $36 e^{x^{2}} x^{4} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}}$ .

$\frac{\frac{2 e^{x^{2}} (22 x {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}}) + 2 e^{x^{2}} (4 {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}}) + 2 e^{x^{2}} (21 x^{3} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}}) + 2 e^{x^{2}} (18 x^{4} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}}) + 70 e^{x^{2}} x^{2} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 2 e^{x^{2}} {(3 x + 1)}^{2} (- 2 x - 1)}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.22.1.5

Factor $2 e^{x^{2}}$ out of $70 e^{x^{2}} x^{2} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}}$ .

$\frac{\frac{2 e^{x^{2}} (22 x {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}}) + 2 e^{x^{2}} (4 {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}}) + 2 e^{x^{2}} (21 x^{3} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}}) + 2 e^{x^{2}} (18 x^{4} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}}) + 2 e^{x^{2}} (35 x^{2} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}}) + 2 e^{x^{2}} {(3 x + 1)}^{2} (- 2 x - 1)}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.22.1.6

Factor $2 e^{x^{2}}$ out of $2 e^{x^{2}} {(3 x + 1)}^{2} (- 2 x - 1)$ .

$\frac{\frac{2 e^{x^{2}} (22 x {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}}) + 2 e^{x^{2}} (4 {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}}) + 2 e^{x^{2}} (21 x^{3} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}}) + 2 e^{x^{2}} (18 x^{4} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}}) + 2 e^{x^{2}} (35 x^{2} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}}) + 2 e^{x^{2}} ({(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.22.1.7

Factor $2 e^{x^{2}}$ out of $2 e^{x^{2}} (22 x {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}}) + 2 e^{x^{2}} (4 {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}})$ .

$\frac{\frac{2 e^{x^{2}} (22 x {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 4 {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}}) + 2 e^{x^{2}} (21 x^{3} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}}) + 2 e^{x^{2}} (18 x^{4} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}}) + 2 e^{x^{2}} (35 x^{2} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}}) + 2 e^{x^{2}} ({(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.22.1.8

Factor $2 e^{x^{2}}$ out of $2 e^{x^{2}} (22 x {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 4 {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}}) + 2 e^{x^{2}} (21 x^{3} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}})$ .

$\frac{\frac{2 e^{x^{2}} (22 x {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 4 {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 21 x^{3} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}}) + 2 e^{x^{2}} (18 x^{4} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}}) + 2 e^{x^{2}} (35 x^{2} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}}) + 2 e^{x^{2}} ({(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.22.1.9

Factor $2 e^{x^{2}}$ out of $2 e^{x^{2}} (22 x {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 4 {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 21 x^{3} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}}) + 2 e^{x^{2}} (18 x^{4} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}})$ .

$\frac{\frac{2 e^{x^{2}} (22 x {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 4 {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 21 x^{3} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 18 x^{4} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}}) + 2 e^{x^{2}} (35 x^{2} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}}) + 2 e^{x^{2}} ({(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.22.1.10

Factor $2 e^{x^{2}}$ out of $2 e^{x^{2}} (22 x {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 4 {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 21 x^{3} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 18 x^{4} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}}) + 2 e^{x^{2}} (35 x^{2} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}})$ .

$\frac{\frac{2 e^{x^{2}} (22 x {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 4 {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 21 x^{3} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 18 x^{4} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 35 x^{2} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}}) + 2 e^{x^{2}} ({(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.22.1.11

Factor $2 e^{x^{2}}$ out of $2 e^{x^{2}} (22 x {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 4 {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 21 x^{3} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 18 x^{4} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 35 x^{2} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}}) + 2 e^{x^{2}} ({(3 x + 1)}^{2} (- 2 x - 1))$ .

$\frac{\frac{2 e^{x^{2}} (22 x {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 4 {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 21 x^{3} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 18 x^{4} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 35 x^{2} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.22.2

Combine exponents.

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Step 4.31.6.22.2.1

Multiply ${(4 x + 1)}^{\frac{1}{2}}$ by ${(4 x + 1)}^{\frac{1}{2}}$ by adding the exponents.

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Step 4.31.6.22.2.1.1

Move ${(4 x + 1)}^{\frac{1}{2}}$ .

$\frac{\frac{2 e^{x^{2}} (22 x ({(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}}) + 4 {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 21 x^{3} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 18 x^{4} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 35 x^{2} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.22.2.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\frac{2 e^{x^{2}} (22 x {(4 x + 1)}^{\frac{1}{2} + \frac{1}{2}} + 4 {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 21 x^{3} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 18 x^{4} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 35 x^{2} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.22.2.1.3

Combine the numerators over the common denominator.

$\frac{\frac{2 e^{x^{2}} (22 x {(4 x + 1)}^{\frac{1 + 1}{2}} + 4 {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 21 x^{3} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 18 x^{4} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 35 x^{2} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.22.2.1.4

Add $1$ and $1$ .

$\frac{\frac{2 e^{x^{2}} (22 x {(4 x + 1)}^{\frac{2}{2}} + 4 {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 21 x^{3} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 18 x^{4} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 35 x^{2} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.22.2.1.5

Divide $2$ by $2$ .

$\frac{\frac{2 e^{x^{2}} (22 x {(4 x + 1)}^{1} + 4 {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 21 x^{3} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 18 x^{4} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 35 x^{2} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.22.2.2

Simplify $22 x {(4 x + 1)}^{1}$ .

$\frac{\frac{2 e^{x^{2}} (22 x (4 x + 1) + 4 {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 21 x^{3} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 18 x^{4} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 35 x^{2} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.22.2.3

Multiply ${(4 x + 1)}^{\frac{1}{2}}$ by ${(4 x + 1)}^{\frac{1}{2}}$ by adding the exponents.

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Step 4.31.6.22.2.3.1

Move ${(4 x + 1)}^{\frac{1}{2}}$ .

$\frac{\frac{2 e^{x^{2}} (22 x (4 x + 1) + 4 ({(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}}) + 21 x^{3} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 18 x^{4} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 35 x^{2} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.22.2.3.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\frac{2 e^{x^{2}} (22 x (4 x + 1) + 4 {(4 x + 1)}^{\frac{1}{2} + \frac{1}{2}} + 21 x^{3} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 18 x^{4} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 35 x^{2} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.22.2.3.3

Combine the numerators over the common denominator.

$\frac{\frac{2 e^{x^{2}} (22 x (4 x + 1) + 4 {(4 x + 1)}^{\frac{1 + 1}{2}} + 21 x^{3} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 18 x^{4} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 35 x^{2} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.22.2.3.4

Add $1$ and $1$ .

$\frac{\frac{2 e^{x^{2}} (22 x (4 x + 1) + 4 {(4 x + 1)}^{\frac{2}{2}} + 21 x^{3} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 18 x^{4} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 35 x^{2} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.22.2.3.5

Divide $2$ by $2$ .

$\frac{\frac{2 e^{x^{2}} (22 x (4 x + 1) + 4 {(4 x + 1)}^{1} + 21 x^{3} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 18 x^{4} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 35 x^{2} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.22.2.4

Simplify $4 {(4 x + 1)}^{1}$ .

$\frac{\frac{2 e^{x^{2}} (22 x (4 x + 1) + 4 (4 x + 1) + 21 x^{3} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 18 x^{4} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 35 x^{2} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.22.2.5

Multiply ${(4 x + 1)}^{\frac{1}{2}}$ by ${(4 x + 1)}^{\frac{1}{2}}$ by adding the exponents.

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Step 4.31.6.22.2.5.1

Move ${(4 x + 1)}^{\frac{1}{2}}$ .

$\frac{\frac{2 e^{x^{2}} (22 x (4 x + 1) + 4 (4 x + 1) + 21 x^{3} ({(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}}) + 18 x^{4} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 35 x^{2} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.22.2.5.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\frac{2 e^{x^{2}} (22 x (4 x + 1) + 4 (4 x + 1) + 21 x^{3} {(4 x + 1)}^{\frac{1}{2} + \frac{1}{2}} + 18 x^{4} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 35 x^{2} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.22.2.5.3

Combine the numerators over the common denominator.

$\frac{\frac{2 e^{x^{2}} (22 x (4 x + 1) + 4 (4 x + 1) + 21 x^{3} {(4 x + 1)}^{\frac{1 + 1}{2}} + 18 x^{4} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 35 x^{2} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.22.2.5.4

Add $1$ and $1$ .

$\frac{\frac{2 e^{x^{2}} (22 x (4 x + 1) + 4 (4 x + 1) + 21 x^{3} {(4 x + 1)}^{\frac{2}{2}} + 18 x^{4} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 35 x^{2} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.22.2.5.5

Divide $2$ by $2$ .

$\frac{\frac{2 e^{x^{2}} (22 x (4 x + 1) + 4 (4 x + 1) + 21 x^{3} {(4 x + 1)}^{1} + 18 x^{4} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 35 x^{2} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.22.2.6

Simplify $21 x^{3} {(4 x + 1)}^{1}$ .

$\frac{\frac{2 e^{x^{2}} (22 x (4 x + 1) + 4 (4 x + 1) + 21 x^{3} (4 x + 1) + 18 x^{4} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + 35 x^{2} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.22.2.7

Multiply ${(4 x + 1)}^{\frac{1}{2}}$ by ${(4 x + 1)}^{\frac{1}{2}}$ by adding the exponents.

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Step 4.31.6.22.2.7.1

Move ${(4 x + 1)}^{\frac{1}{2}}$ .

$\frac{\frac{2 e^{x^{2}} (22 x (4 x + 1) + 4 (4 x + 1) + 21 x^{3} (4 x + 1) + 18 x^{4} ({(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}}) + 35 x^{2} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.22.2.7.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\frac{2 e^{x^{2}} (22 x (4 x + 1) + 4 (4 x + 1) + 21 x^{3} (4 x + 1) + 18 x^{4} {(4 x + 1)}^{\frac{1}{2} + \frac{1}{2}} + 35 x^{2} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.22.2.7.3

Combine the numerators over the common denominator.

$\frac{\frac{2 e^{x^{2}} (22 x (4 x + 1) + 4 (4 x + 1) + 21 x^{3} (4 x + 1) + 18 x^{4} {(4 x + 1)}^{\frac{1 + 1}{2}} + 35 x^{2} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.22.2.7.4

Add $1$ and $1$ .

$\frac{\frac{2 e^{x^{2}} (22 x (4 x + 1) + 4 (4 x + 1) + 21 x^{3} (4 x + 1) + 18 x^{4} {(4 x + 1)}^{\frac{2}{2}} + 35 x^{2} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.22.2.7.5

Divide $2$ by $2$ .

$\frac{\frac{2 e^{x^{2}} (22 x (4 x + 1) + 4 (4 x + 1) + 21 x^{3} (4 x + 1) + 18 x^{4} {(4 x + 1)}^{1} + 35 x^{2} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.22.2.8

Simplify $18 x^{4} {(4 x + 1)}^{1}$ .

$\frac{\frac{2 e^{x^{2}} (22 x (4 x + 1) + 4 (4 x + 1) + 21 x^{3} (4 x + 1) + 18 x^{4} (4 x + 1) + 35 x^{2} {(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.22.2.9

Multiply ${(4 x + 1)}^{\frac{1}{2}}$ by ${(4 x + 1)}^{\frac{1}{2}}$ by adding the exponents.

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Step 4.31.6.22.2.9.1

Move ${(4 x + 1)}^{\frac{1}{2}}$ .

$\frac{\frac{2 e^{x^{2}} (22 x (4 x + 1) + 4 (4 x + 1) + 21 x^{3} (4 x + 1) + 18 x^{4} (4 x + 1) + 35 x^{2} ({(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}}) + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.22.2.9.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\frac{2 e^{x^{2}} (22 x (4 x + 1) + 4 (4 x + 1) + 21 x^{3} (4 x + 1) + 18 x^{4} (4 x + 1) + 35 x^{2} {(4 x + 1)}^{\frac{1}{2} + \frac{1}{2}} + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.22.2.9.3

Combine the numerators over the common denominator.

$\frac{\frac{2 e^{x^{2}} (22 x (4 x + 1) + 4 (4 x + 1) + 21 x^{3} (4 x + 1) + 18 x^{4} (4 x + 1) + 35 x^{2} {(4 x + 1)}^{\frac{1 + 1}{2}} + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.22.2.9.4

Add $1$ and $1$ .

$\frac{\frac{2 e^{x^{2}} (22 x (4 x + 1) + 4 (4 x + 1) + 21 x^{3} (4 x + 1) + 18 x^{4} (4 x + 1) + 35 x^{2} {(4 x + 1)}^{\frac{2}{2}} + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.22.2.9.5

Divide $2$ by $2$ .

$\frac{\frac{2 e^{x^{2}} (22 x (4 x + 1) + 4 (4 x + 1) + 21 x^{3} (4 x + 1) + 18 x^{4} (4 x + 1) + 35 x^{2} {(4 x + 1)}^{1} + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.22.2.10

Simplify $35 x^{2} {(4 x + 1)}^{1}$ .

$\frac{\frac{2 e^{x^{2}} (22 x (4 x + 1) + 4 (4 x + 1) + 21 x^{3} (4 x + 1) + 18 x^{4} (4 x + 1) + 35 x^{2} (4 x + 1) + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23

Simplify the numerator.

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Step 4.31.6.23.1

Apply the distributive property.

$\frac{\frac{2 e^{x^{2}} (22 x (4 x) + 22 x \cdot 1 + 4 (4 x + 1) + 21 x^{3} (4 x + 1) + 18 x^{4} (4 x + 1) + 35 x^{2} (4 x + 1) + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.2

Rewrite using the commutative property of multiplication.

$\frac{\frac{2 e^{x^{2}} (22 \cdot 4 x \cdot x + 22 x \cdot 1 + 4 (4 x + 1) + 21 x^{3} (4 x + 1) + 18 x^{4} (4 x + 1) + 35 x^{2} (4 x + 1) + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.3

Multiply $22$ by $1$ .

$\frac{\frac{2 e^{x^{2}} (22 \cdot 4 x \cdot x + 22 x + 4 (4 x + 1) + 21 x^{3} (4 x + 1) + 18 x^{4} (4 x + 1) + 35 x^{2} (4 x + 1) + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.4

Simplify each term.

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Step 4.31.6.23.4.1

Multiply $x$ by $x$ by adding the exponents.

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Step 4.31.6.23.4.1.1

Move $x$ .

$\frac{\frac{2 e^{x^{2}} (22 \cdot 4 (x \cdot x) + 22 x + 4 (4 x + 1) + 21 x^{3} (4 x + 1) + 18 x^{4} (4 x + 1) + 35 x^{2} (4 x + 1) + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.4.1.2

Multiply $x$ by $x$ .

$\frac{\frac{2 e^{x^{2}} (22 \cdot 4 x^{2} + 22 x + 4 (4 x + 1) + 21 x^{3} (4 x + 1) + 18 x^{4} (4 x + 1) + 35 x^{2} (4 x + 1) + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.4.2

Multiply $22$ by $4$ .

$\frac{\frac{2 e^{x^{2}} (88 x^{2} + 22 x + 4 (4 x + 1) + 21 x^{3} (4 x + 1) + 18 x^{4} (4 x + 1) + 35 x^{2} (4 x + 1) + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.5

Apply the distributive property.

$\frac{\frac{2 e^{x^{2}} (88 x^{2} + 22 x + 4 (4 x) + 4 \cdot 1 + 21 x^{3} (4 x + 1) + 18 x^{4} (4 x + 1) + 35 x^{2} (4 x + 1) + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.6

Multiply $4$ by $4$ .

$\frac{\frac{2 e^{x^{2}} (88 x^{2} + 22 x + 16 x + 4 \cdot 1 + 21 x^{3} (4 x + 1) + 18 x^{4} (4 x + 1) + 35 x^{2} (4 x + 1) + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.7

Multiply $4$ by $1$ .

$\frac{\frac{2 e^{x^{2}} (88 x^{2} + 22 x + 16 x + 4 + 21 x^{3} (4 x + 1) + 18 x^{4} (4 x + 1) + 35 x^{2} (4 x + 1) + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.8

Apply the distributive property.

$\frac{\frac{2 e^{x^{2}} (88 x^{2} + 22 x + 16 x + 4 + 21 x^{3} (4 x) + 21 x^{3} \cdot 1 + 18 x^{4} (4 x + 1) + 35 x^{2} (4 x + 1) + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.9

Rewrite using the commutative property of multiplication.

$\frac{\frac{2 e^{x^{2}} (88 x^{2} + 22 x + 16 x + 4 + 21 \cdot 4 x^{3} x + 21 x^{3} \cdot 1 + 18 x^{4} (4 x + 1) + 35 x^{2} (4 x + 1) + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.10

Multiply $21$ by $1$ .

$\frac{\frac{2 e^{x^{2}} (88 x^{2} + 22 x + 16 x + 4 + 21 \cdot 4 x^{3} x + 21 x^{3} + 18 x^{4} (4 x + 1) + 35 x^{2} (4 x + 1) + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.11

Simplify each term.

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Step 4.31.6.23.11.1

Multiply $x^{3}$ by $x$ by adding the exponents.

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Step 4.31.6.23.11.1.1

Move $x$ .

$\frac{\frac{2 e^{x^{2}} (88 x^{2} + 22 x + 16 x + 4 + 21 \cdot 4 (x \cdot x^{3}) + 21 x^{3} + 18 x^{4} (4 x + 1) + 35 x^{2} (4 x + 1) + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.11.1.2

Multiply $x$ by $x^{3}$ .

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Step 4.31.6.23.11.1.2.1

Raise $x$ to the power of $1$ .

$\frac{\frac{2 e^{x^{2}} (88 x^{2} + 22 x + 16 x + 4 + 21 \cdot 4 (x^{1} x^{3}) + 21 x^{3} + 18 x^{4} (4 x + 1) + 35 x^{2} (4 x + 1) + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.11.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\frac{2 e^{x^{2}} (88 x^{2} + 22 x + 16 x + 4 + 21 \cdot 4 x^{1 + 3} + 21 x^{3} + 18 x^{4} (4 x + 1) + 35 x^{2} (4 x + 1) + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.11.1.3

Add $1$ and $3$ .

$\frac{\frac{2 e^{x^{2}} (88 x^{2} + 22 x + 16 x + 4 + 21 \cdot 4 x^{4} + 21 x^{3} + 18 x^{4} (4 x + 1) + 35 x^{2} (4 x + 1) + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.11.2

Multiply $21$ by $4$ .

$\frac{\frac{2 e^{x^{2}} (88 x^{2} + 22 x + 16 x + 4 + 84 x^{4} + 21 x^{3} + 18 x^{4} (4 x + 1) + 35 x^{2} (4 x + 1) + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.12

Apply the distributive property.

$\frac{\frac{2 e^{x^{2}} (88 x^{2} + 22 x + 16 x + 4 + 84 x^{4} + 21 x^{3} + 18 x^{4} (4 x) + 18 x^{4} \cdot 1 + 35 x^{2} (4 x + 1) + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.13

Rewrite using the commutative property of multiplication.

$\frac{\frac{2 e^{x^{2}} (88 x^{2} + 22 x + 16 x + 4 + 84 x^{4} + 21 x^{3} + 18 \cdot 4 x^{4} x + 18 x^{4} \cdot 1 + 35 x^{2} (4 x + 1) + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.14

Multiply $18$ by $1$ .

$\frac{\frac{2 e^{x^{2}} (88 x^{2} + 22 x + 16 x + 4 + 84 x^{4} + 21 x^{3} + 18 \cdot 4 x^{4} x + 18 x^{4} + 35 x^{2} (4 x + 1) + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.15

Simplify each term.

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Step 4.31.6.23.15.1

Multiply $x^{4}$ by $x$ by adding the exponents.

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Step 4.31.6.23.15.1.1

Move $x$ .

$\frac{\frac{2 e^{x^{2}} (88 x^{2} + 22 x + 16 x + 4 + 84 x^{4} + 21 x^{3} + 18 \cdot 4 (x \cdot x^{4}) + 18 x^{4} + 35 x^{2} (4 x + 1) + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.15.1.2

Multiply $x$ by $x^{4}$ .

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Step 4.31.6.23.15.1.2.1

Raise $x$ to the power of $1$ .

$\frac{\frac{2 e^{x^{2}} (88 x^{2} + 22 x + 16 x + 4 + 84 x^{4} + 21 x^{3} + 18 \cdot 4 (x^{1} x^{4}) + 18 x^{4} + 35 x^{2} (4 x + 1) + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.15.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\frac{2 e^{x^{2}} (88 x^{2} + 22 x + 16 x + 4 + 84 x^{4} + 21 x^{3} + 18 \cdot 4 x^{1 + 4} + 18 x^{4} + 35 x^{2} (4 x + 1) + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.15.1.3

Add $1$ and $4$ .

$\frac{\frac{2 e^{x^{2}} (88 x^{2} + 22 x + 16 x + 4 + 84 x^{4} + 21 x^{3} + 18 \cdot 4 x^{5} + 18 x^{4} + 35 x^{2} (4 x + 1) + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.15.2

Multiply $18$ by $4$ .

$\frac{\frac{2 e^{x^{2}} (88 x^{2} + 22 x + 16 x + 4 + 84 x^{4} + 21 x^{3} + 72 x^{5} + 18 x^{4} + 35 x^{2} (4 x + 1) + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.16

Apply the distributive property.

$\frac{\frac{2 e^{x^{2}} (88 x^{2} + 22 x + 16 x + 4 + 84 x^{4} + 21 x^{3} + 72 x^{5} + 18 x^{4} + 35 x^{2} (4 x) + 35 x^{2} \cdot 1 + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.17

Rewrite using the commutative property of multiplication.

$\frac{\frac{2 e^{x^{2}} (88 x^{2} + 22 x + 16 x + 4 + 84 x^{4} + 21 x^{3} + 72 x^{5} + 18 x^{4} + 35 \cdot 4 x^{2} x + 35 x^{2} \cdot 1 + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.18

Multiply $35$ by $1$ .

$\frac{\frac{2 e^{x^{2}} (88 x^{2} + 22 x + 16 x + 4 + 84 x^{4} + 21 x^{3} + 72 x^{5} + 18 x^{4} + 35 \cdot 4 x^{2} x + 35 x^{2} + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.19

Simplify each term.

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Step 4.31.6.23.19.1

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 4.31.6.23.19.1.1

Move $x$ .

$\frac{\frac{2 e^{x^{2}} (88 x^{2} + 22 x + 16 x + 4 + 84 x^{4} + 21 x^{3} + 72 x^{5} + 18 x^{4} + 35 \cdot 4 (x \cdot x^{2}) + 35 x^{2} + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.19.1.2

Multiply $x$ by $x^{2}$ .

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Step 4.31.6.23.19.1.2.1

Raise $x$ to the power of $1$ .

$\frac{\frac{2 e^{x^{2}} (88 x^{2} + 22 x + 16 x + 4 + 84 x^{4} + 21 x^{3} + 72 x^{5} + 18 x^{4} + 35 \cdot 4 (x^{1} x^{2}) + 35 x^{2} + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.19.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\frac{2 e^{x^{2}} (88 x^{2} + 22 x + 16 x + 4 + 84 x^{4} + 21 x^{3} + 72 x^{5} + 18 x^{4} + 35 \cdot 4 x^{1 + 2} + 35 x^{2} + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.19.1.3

Add $1$ and $2$ .

$\frac{\frac{2 e^{x^{2}} (88 x^{2} + 22 x + 16 x + 4 + 84 x^{4} + 21 x^{3} + 72 x^{5} + 18 x^{4} + 35 \cdot 4 x^{3} + 35 x^{2} + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.19.2

Multiply $35$ by $4$ .

$\frac{\frac{2 e^{x^{2}} (88 x^{2} + 22 x + 16 x + 4 + 84 x^{4} + 21 x^{3} + 72 x^{5} + 18 x^{4} + 140 x^{3} + 35 x^{2} + {(3 x + 1)}^{2} (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.20

Rewrite ${(3 x + 1)}^{2}$ as $(3 x + 1) (3 x + 1)$ .

$\frac{\frac{2 e^{x^{2}} (88 x^{2} + 22 x + 16 x + 4 + 84 x^{4} + 21 x^{3} + 72 x^{5} + 18 x^{4} + 140 x^{3} + 35 x^{2} + (3 x + 1) (3 x + 1) (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.21

Expand $(3 x + 1) (3 x + 1)$ using the FOIL Method.

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Step 4.31.6.23.21.1

Apply the distributive property.

$\frac{\frac{2 e^{x^{2}} (88 x^{2} + 22 x + 16 x + 4 + 84 x^{4} + 21 x^{3} + 72 x^{5} + 18 x^{4} + 140 x^{3} + 35 x^{2} + (3 x (3 x + 1) + 1 (3 x + 1)) (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.21.2

Apply the distributive property.

$\frac{\frac{2 e^{x^{2}} (88 x^{2} + 22 x + 16 x + 4 + 84 x^{4} + 21 x^{3} + 72 x^{5} + 18 x^{4} + 140 x^{3} + 35 x^{2} + (3 x (3 x) + 3 x \cdot 1 + 1 (3 x + 1)) (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.21.3

Apply the distributive property.

$\frac{\frac{2 e^{x^{2}} (88 x^{2} + 22 x + 16 x + 4 + 84 x^{4} + 21 x^{3} + 72 x^{5} + 18 x^{4} + 140 x^{3} + 35 x^{2} + (3 x (3 x) + 3 x \cdot 1 + 1 (3 x) + 1 \cdot 1) (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.22

Simplify and combine like terms.

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Step 4.31.6.23.22.1

Simplify each term.

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Step 4.31.6.23.22.1.1

Rewrite using the commutative property of multiplication.

$\frac{\frac{2 e^{x^{2}} (88 x^{2} + 22 x + 16 x + 4 + 84 x^{4} + 21 x^{3} + 72 x^{5} + 18 x^{4} + 140 x^{3} + 35 x^{2} + (3 \cdot 3 x \cdot x + 3 x \cdot 1 + 1 (3 x) + 1 \cdot 1) (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.22.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 4.31.6.23.22.1.2.1

Move $x$ .

$\frac{\frac{2 e^{x^{2}} (88 x^{2} + 22 x + 16 x + 4 + 84 x^{4} + 21 x^{3} + 72 x^{5} + 18 x^{4} + 140 x^{3} + 35 x^{2} + (3 \cdot 3 (x \cdot x) + 3 x \cdot 1 + 1 (3 x) + 1 \cdot 1) (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.22.1.2.2

Multiply $x$ by $x$ .

$\frac{\frac{2 e^{x^{2}} (88 x^{2} + 22 x + 16 x + 4 + 84 x^{4} + 21 x^{3} + 72 x^{5} + 18 x^{4} + 140 x^{3} + 35 x^{2} + (3 \cdot 3 x^{2} + 3 x \cdot 1 + 1 (3 x) + 1 \cdot 1) (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.22.1.3

Multiply $3$ by $3$ .

$\frac{\frac{2 e^{x^{2}} (88 x^{2} + 22 x + 16 x + 4 + 84 x^{4} + 21 x^{3} + 72 x^{5} + 18 x^{4} + 140 x^{3} + 35 x^{2} + (9 x^{2} + 3 x \cdot 1 + 1 (3 x) + 1 \cdot 1) (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.22.1.4

Multiply $3$ by $1$ .

$\frac{\frac{2 e^{x^{2}} (88 x^{2} + 22 x + 16 x + 4 + 84 x^{4} + 21 x^{3} + 72 x^{5} + 18 x^{4} + 140 x^{3} + 35 x^{2} + (9 x^{2} + 3 x + 1 (3 x) + 1 \cdot 1) (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.22.1.5

Multiply $3 x$ by $1$ .

$\frac{\frac{2 e^{x^{2}} (88 x^{2} + 22 x + 16 x + 4 + 84 x^{4} + 21 x^{3} + 72 x^{5} + 18 x^{4} + 140 x^{3} + 35 x^{2} + (9 x^{2} + 3 x + 3 x + 1 \cdot 1) (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.22.1.6

Multiply $1$ by $1$ .

$\frac{\frac{2 e^{x^{2}} (88 x^{2} + 22 x + 16 x + 4 + 84 x^{4} + 21 x^{3} + 72 x^{5} + 18 x^{4} + 140 x^{3} + 35 x^{2} + (9 x^{2} + 3 x + 3 x + 1) (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.22.2

Add $3 x$ and $3 x$ .

$\frac{\frac{2 e^{x^{2}} (88 x^{2} + 22 x + 16 x + 4 + 84 x^{4} + 21 x^{3} + 72 x^{5} + 18 x^{4} + 140 x^{3} + 35 x^{2} + (9 x^{2} + 6 x + 1) (- 2 x - 1))}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.23

Expand $(9 x^{2} + 6 x + 1) (- 2 x - 1)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{\frac{2 e^{x^{2}} (88 x^{2} + 22 x + 16 x + 4 + 84 x^{4} + 21 x^{3} + 72 x^{5} + 18 x^{4} + 140 x^{3} + 35 x^{2} + 9 x^{2} (- 2 x) + 9 x^{2} \cdot - 1 + 6 x (- 2 x) + 6 x \cdot - 1 + 1 (- 2 x) + 1 \cdot - 1)}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.24

Simplify each term.

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Step 4.31.6.23.24.1

Rewrite using the commutative property of multiplication.

$\frac{\frac{2 e^{x^{2}} (88 x^{2} + 22 x + 16 x + 4 + 84 x^{4} + 21 x^{3} + 72 x^{5} + 18 x^{4} + 140 x^{3} + 35 x^{2} + 9 \cdot - 2 x^{2} x + 9 x^{2} \cdot - 1 + 6 x (- 2 x) + 6 x \cdot - 1 + 1 (- 2 x) + 1 \cdot - 1)}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.24.2

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 4.31.6.23.24.2.1

Move $x$ .

$\frac{\frac{2 e^{x^{2}} (88 x^{2} + 22 x + 16 x + 4 + 84 x^{4} + 21 x^{3} + 72 x^{5} + 18 x^{4} + 140 x^{3} + 35 x^{2} + 9 \cdot - 2 (x \cdot x^{2}) + 9 x^{2} \cdot - 1 + 6 x (- 2 x) + 6 x \cdot - 1 + 1 (- 2 x) + 1 \cdot - 1)}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.24.2.2

Multiply $x$ by $x^{2}$ .

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Step 4.31.6.23.24.2.2.1

Raise $x$ to the power of $1$ .

$\frac{\frac{2 e^{x^{2}} (88 x^{2} + 22 x + 16 x + 4 + 84 x^{4} + 21 x^{3} + 72 x^{5} + 18 x^{4} + 140 x^{3} + 35 x^{2} + 9 \cdot - 2 (x^{1} x^{2}) + 9 x^{2} \cdot - 1 + 6 x (- 2 x) + 6 x \cdot - 1 + 1 (- 2 x) + 1 \cdot - 1)}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.24.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\frac{2 e^{x^{2}} (88 x^{2} + 22 x + 16 x + 4 + 84 x^{4} + 21 x^{3} + 72 x^{5} + 18 x^{4} + 140 x^{3} + 35 x^{2} + 9 \cdot - 2 x^{1 + 2} + 9 x^{2} \cdot - 1 + 6 x (- 2 x) + 6 x \cdot - 1 + 1 (- 2 x) + 1 \cdot - 1)}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.24.2.3

Add $1$ and $2$ .

$\frac{\frac{2 e^{x^{2}} (88 x^{2} + 22 x + 16 x + 4 + 84 x^{4} + 21 x^{3} + 72 x^{5} + 18 x^{4} + 140 x^{3} + 35 x^{2} + 9 \cdot - 2 x^{3} + 9 x^{2} \cdot - 1 + 6 x (- 2 x) + 6 x \cdot - 1 + 1 (- 2 x) + 1 \cdot - 1)}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.24.3

Multiply $9$ by $- 2$ .

$\frac{\frac{2 e^{x^{2}} (88 x^{2} + 22 x + 16 x + 4 + 84 x^{4} + 21 x^{3} + 72 x^{5} + 18 x^{4} + 140 x^{3} + 35 x^{2} - 18 x^{3} + 9 x^{2} \cdot - 1 + 6 x (- 2 x) + 6 x \cdot - 1 + 1 (- 2 x) + 1 \cdot - 1)}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.24.4

Multiply $- 1$ by $9$ .

$\frac{\frac{2 e^{x^{2}} (88 x^{2} + 22 x + 16 x + 4 + 84 x^{4} + 21 x^{3} + 72 x^{5} + 18 x^{4} + 140 x^{3} + 35 x^{2} - 18 x^{3} - 9 x^{2} + 6 x (- 2 x) + 6 x \cdot - 1 + 1 (- 2 x) + 1 \cdot - 1)}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.24.5

Rewrite using the commutative property of multiplication.

$\frac{\frac{2 e^{x^{2}} (88 x^{2} + 22 x + 16 x + 4 + 84 x^{4} + 21 x^{3} + 72 x^{5} + 18 x^{4} + 140 x^{3} + 35 x^{2} - 18 x^{3} - 9 x^{2} + 6 \cdot - 2 x \cdot x + 6 x \cdot - 1 + 1 (- 2 x) + 1 \cdot - 1)}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.24.6

Multiply $x$ by $x$ by adding the exponents.

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Step 4.31.6.23.24.6.1

Move $x$ .

$\frac{\frac{2 e^{x^{2}} (88 x^{2} + 22 x + 16 x + 4 + 84 x^{4} + 21 x^{3} + 72 x^{5} + 18 x^{4} + 140 x^{3} + 35 x^{2} - 18 x^{3} - 9 x^{2} + 6 \cdot - 2 (x \cdot x) + 6 x \cdot - 1 + 1 (- 2 x) + 1 \cdot - 1)}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.24.6.2

Multiply $x$ by $x$ .

$\frac{\frac{2 e^{x^{2}} (88 x^{2} + 22 x + 16 x + 4 + 84 x^{4} + 21 x^{3} + 72 x^{5} + 18 x^{4} + 140 x^{3} + 35 x^{2} - 18 x^{3} - 9 x^{2} + 6 \cdot - 2 x^{2} + 6 x \cdot - 1 + 1 (- 2 x) + 1 \cdot - 1)}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.24.7

Multiply $6$ by $- 2$ .

$\frac{\frac{2 e^{x^{2}} (88 x^{2} + 22 x + 16 x + 4 + 84 x^{4} + 21 x^{3} + 72 x^{5} + 18 x^{4} + 140 x^{3} + 35 x^{2} - 18 x^{3} - 9 x^{2} - 12 x^{2} + 6 x \cdot - 1 + 1 (- 2 x) + 1 \cdot - 1)}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.24.8

Multiply $- 1$ by $6$ .

$\frac{\frac{2 e^{x^{2}} (88 x^{2} + 22 x + 16 x + 4 + 84 x^{4} + 21 x^{3} + 72 x^{5} + 18 x^{4} + 140 x^{3} + 35 x^{2} - 18 x^{3} - 9 x^{2} - 12 x^{2} - 6 x + 1 (- 2 x) + 1 \cdot - 1)}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.24.9

Multiply $- 2 x$ by $1$ .

$\frac{\frac{2 e^{x^{2}} (88 x^{2} + 22 x + 16 x + 4 + 84 x^{4} + 21 x^{3} + 72 x^{5} + 18 x^{4} + 140 x^{3} + 35 x^{2} - 18 x^{3} - 9 x^{2} - 12 x^{2} - 6 x - 2 x + 1 \cdot - 1)}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.24.10

Multiply $- 1$ by $1$ .

$\frac{\frac{2 e^{x^{2}} (88 x^{2} + 22 x + 16 x + 4 + 84 x^{4} + 21 x^{3} + 72 x^{5} + 18 x^{4} + 140 x^{3} + 35 x^{2} - 18 x^{3} - 9 x^{2} - 12 x^{2} - 6 x - 2 x - 1)}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.25

Subtract $12 x^{2}$ from $- 9 x^{2}$ .

$\frac{\frac{2 e^{x^{2}} (88 x^{2} + 22 x + 16 x + 4 + 84 x^{4} + 21 x^{3} + 72 x^{5} + 18 x^{4} + 140 x^{3} + 35 x^{2} - 18 x^{3} - 21 x^{2} - 6 x - 2 x - 1)}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.26

Subtract $2 x$ from $- 6 x$ .

$\frac{\frac{2 e^{x^{2}} (88 x^{2} + 22 x + 16 x + 4 + 84 x^{4} + 21 x^{3} + 72 x^{5} + 18 x^{4} + 140 x^{3} + 35 x^{2} - 18 x^{3} - 21 x^{2} - 8 x - 1)}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.27

Add $88 x^{2}$ and $35 x^{2}$ .

$\frac{\frac{2 e^{x^{2}} (22 x + 16 x + 4 + 84 x^{4} + 21 x^{3} + 72 x^{5} + 18 x^{4} + 140 x^{3} + 123 x^{2} - 18 x^{3} - 21 x^{2} - 8 x - 1)}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.28

Add $22 x$ and $16 x$ .

$\frac{\frac{2 e^{x^{2}} (38 x + 4 + 84 x^{4} + 21 x^{3} + 72 x^{5} + 18 x^{4} + 140 x^{3} + 123 x^{2} - 18 x^{3} - 21 x^{2} - 8 x - 1)}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.29

Subtract $8 x$ from $38 x$ .

$\frac{\frac{2 e^{x^{2}} (4 + 84 x^{4} + 21 x^{3} + 72 x^{5} + 18 x^{4} + 140 x^{3} + 123 x^{2} - 18 x^{3} - 21 x^{2} + 30 x - 1)}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.30

Subtract $1$ from $4$ .

$\frac{\frac{2 e^{x^{2}} (84 x^{4} + 21 x^{3} + 72 x^{5} + 18 x^{4} + 140 x^{3} + 123 x^{2} - 18 x^{3} - 21 x^{2} + 30 x + 3)}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.31

Add $84 x^{4}$ and $18 x^{4}$ .

$\frac{\frac{2 e^{x^{2}} (102 x^{4} + 21 x^{3} + 72 x^{5} + 140 x^{3} + 123 x^{2} - 18 x^{3} - 21 x^{2} + 30 x + 3)}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.32

Add $21 x^{3}$ and $140 x^{3}$ .

$\frac{\frac{2 e^{x^{2}} (102 x^{4} + 72 x^{5} + 161 x^{3} + 123 x^{2} - 18 x^{3} - 21 x^{2} + 30 x + 3)}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.33

Subtract $18 x^{3}$ from $161 x^{3}$ .

$\frac{\frac{2 e^{x^{2}} (102 x^{4} + 72 x^{5} + 143 x^{3} + 123 x^{2} - 21 x^{2} + 30 x + 3)}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.34

Subtract $21 x^{2}$ from $123 x^{2}$ .

$\frac{\frac{2 e^{x^{2}} (102 x^{4} + 72 x^{5} + 143 x^{3} + 102 x^{2} + 30 x + 3)}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.6.23.35

Reorder terms.

$\frac{\frac{2 e^{x^{2}} (72 x^{5} + 102 x^{4} + 143 x^{3} + 102 x^{2} + 30 x + 3)}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.7

Combine terms.

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Step 4.31.7.1

Multiply $e^{x^{2}}$ by $1$ .

$\frac{\frac{2 e^{x^{2}} (72 x^{5} + 102 x^{4} + 143 x^{3} + 102 x^{2} + 30 x + 3)}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}}) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.7.2

Rewrite $\frac{\frac{2 e^{x^{2}} (72 x^{5} + 102 x^{4} + 143 x^{3} + 102 x^{2} + 30 x + 3)}{{(4 x + 1)}^{\frac{1}{2}}}}{(e^{x^{2}} (2 x) + e^{x^{2}}) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$ as a product.

$\frac{2 e^{x^{2}} (72 x^{5} + 102 x^{4} + 143 x^{3} + 102 x^{2} + 30 x + 3)}{{(4 x + 1)}^{\frac{1}{2}}} \cdot \frac{1}{(e^{x^{2}} (2 x) + e^{x^{2}}) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$

Step 4.31.7.3

Multiply $\frac{2 e^{x^{2}} (72 x^{5} + 102 x^{4} + 143 x^{3} + 102 x^{2} + 30 x + 3)}{{(4 x + 1)}^{\frac{1}{2}}}$ by $\frac{1}{(e^{x^{2}} (2 x) + e^{x^{2}}) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}}}$ .

$\frac{2 e^{x^{2}} (72 x^{5} + 102 x^{4} + 143 x^{3} + 102 x^{2} + 30 x + 3)}{{(4 x + 1)}^{\frac{1}{2}} ((e^{x^{2}} (2 x) + e^{x^{2}}) {(3 x + 1)}^{2} {(4 x + 1)}^{\frac{1}{2}})}$

Step 4.31.7.4

Multiply ${(4 x + 1)}^{\frac{1}{2}}$ by ${(4 x + 1)}^{\frac{1}{2}}$ by adding the exponents.

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Step 4.31.7.4.1

Move ${(4 x + 1)}^{\frac{1}{2}}$ .

$\frac{2 e^{x^{2}} (72 x^{5} + 102 x^{4} + 143 x^{3} + 102 x^{2} + 30 x + 3)}{{(4 x + 1)}^{\frac{1}{2}} {(4 x + 1)}^{\frac{1}{2}} ((e^{x^{2}} (2 x) + e^{x^{2}}) {(3 x + 1)}^{2})}$

Step 4.31.7.4.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{2 e^{x^{2}} (72 x^{5} + 102 x^{4} + 143 x^{3} + 102 x^{2} + 30 x + 3)}{{(4 x + 1)}^{\frac{1}{2} + \frac{1}{2}} ((e^{x^{2}} (2 x) + e^{x^{2}}) {(3 x + 1)}^{2})}$

Step 4.31.7.4.3

Combine the numerators over the common denominator.

$\frac{2 e^{x^{2}} (72 x^{5} + 102 x^{4} + 143 x^{3} + 102 x^{2} + 30 x + 3)}{{(4 x + 1)}^{\frac{1 + 1}{2}} ((e^{x^{2}} (2 x) + e^{x^{2}}) {(3 x + 1)}^{2})}$

Step 4.31.7.4.4

Add $1$ and $1$ .

$\frac{2 e^{x^{2}} (72 x^{5} + 102 x^{4} + 143 x^{3} + 102 x^{2} + 30 x + 3)}{{(4 x + 1)}^{\frac{2}{2}} ((e^{x^{2}} (2 x) + e^{x^{2}}) {(3 x + 1)}^{2})}$

Step 4.31.7.4.5

Divide $2$ by $2$ .

$\frac{2 e^{x^{2}} (72 x^{5} + 102 x^{4} + 143 x^{3} + 102 x^{2} + 30 x + 3)}{{(4 x + 1)}^{1} ((e^{x^{2}} (2 x) + e^{x^{2}}) {(3 x + 1)}^{2})}$

Step 4.31.7.5

Simplify ${(4 x + 1)}^{1} ((e^{x^{2}} (2 x) + e^{x^{2}}) {(3 x + 1)}^{2})$ .

$\frac{2 e^{x^{2}} (72 x^{5} + 102 x^{4} + 143 x^{3} + 102 x^{2} + 30 x + 3)}{(4 x + 1) (e^{x^{2}} (2 x) + e^{x^{2}}) {(3 x + 1)}^{2}}$

Step 4.31.8

Reorder terms.

$\frac{2 e^{x^{2}} (72 x^{5} + 102 x^{4} + 143 x^{3} + 102 x^{2} + 30 x + 3)}{{(3 x + 1)}^{2} (4 x + 1) (e^{x^{2}} (2 x) + e^{x^{2}})}$

Step 4.31.9

Factor $e^{x^{2}}$ out of $e^{x^{2}} (2 x) + e^{x^{2}}$ .

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Step 4.31.9.1

Multiply by $1$ .

$\frac{2 e^{x^{2}} (72 x^{5} + 102 x^{4} + 143 x^{3} + 102 x^{2} + 30 x + 3)}{{(3 x + 1)}^{2} (4 x + 1) (e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1)}$

Step 4.31.9.2

Factor $e^{x^{2}}$ out of $e^{x^{2}} (2 x) + e^{x^{2}} \cdot 1$ .

$\frac{2 e^{x^{2}} (72 x^{5} + 102 x^{4} + 143 x^{3} + 102 x^{2} + 30 x + 3)}{{(3 x + 1)}^{2} (4 x + 1) (e^{x^{2}} (2 x + 1))}$

$\frac{2 e^{x^{2}} (72 x^{5} + 102 x^{4} + 143 x^{3} + 102 x^{2} + 30 x + 3)}{{(3 x + 1)}^{2} (4 x + 1) e^{x^{2}} (2 x + 1)}$

Step 4.31.10

Cancel the common factor of $e^{x^{2}}$ .

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Step 4.31.10.1

Cancel the common factor.

$\frac{2 e^{x^{2}} (72 x^{5} + 102 x^{4} + 143 x^{3} + 102 x^{2} + 30 x + 3)}{{(3 x + 1)}^{2} (4 x + 1) e^{x^{2}} (2 x + 1)}$

Step 4.31.10.2

Rewrite the expression.

$\frac{2 (72 x^{5} + 102 x^{4} + 143 x^{3} + 102 x^{2} + 30 x + 3)}{{(3 x + 1)}^{2} (4 x + 1) (2 x + 1)}$

Step 5

Reform the equation by setting the left side equal to the right side.

$y' = \frac{2 (72 x^{5} + 102 x^{4} + 143 x^{3} + 102 x^{2} + 30 x + 3)}{{(3 x + 1)}^{2} (4 x + 1) (2 x + 1)}$

Step 6

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = \frac{2 (72 x^{5} + 102 x^{4} + 143 x^{3} + 102 x^{2} + 30 x + 3)}{{(3 x + 1)}^{2} (4 x + 1) (2 x + 1)}$