Calculus Examples

$y = \sqrt{x} e^{x^{2}} {(x^{2} + 1)}^{10}$

Step 1

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

$\frac{d}{d x} [x^{\frac{1}{2}} e^{x^{2}} {(x^{2} + 1)}^{10}]$

Step 2

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 $f (x) = x^{\frac{1}{2}} e^{x^{2}}$ and $g (x) = {(x^{2} + 1)}^{10}$ .

$x^{\frac{1}{2}} e^{x^{2}} \frac{d}{d x} [{(x^{2} + 1)}^{10}] + {(x^{2} + 1)}^{10} \frac{d}{d x} [x^{\frac{1}{2}} e^{x^{2}}]$

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{10}$ and $g (x) = x^{2} + 1$ .

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Step 3.1

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

$x^{\frac{1}{2}} e^{x^{2}} (\frac{d}{d u_{1}} [{u_{1}}^{10}] \frac{d}{d x} [x^{2} + 1]) + {(x^{2} + 1)}^{10} \frac{d}{d x} [x^{\frac{1}{2}} e^{x^{2}}]$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 10$ .

$x^{\frac{1}{2}} e^{x^{2}} (10 {u_{1}}^{9} \frac{d}{d x} [x^{2} + 1]) + {(x^{2} + 1)}^{10} \frac{d}{d x} [x^{\frac{1}{2}} e^{x^{2}}]$

Step 3.3

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

$x^{\frac{1}{2}} e^{x^{2}} (10 {(x^{2} + 1)}^{9} \frac{d}{d x} [x^{2} + 1]) + {(x^{2} + 1)}^{10} \frac{d}{d x} [x^{\frac{1}{2}} e^{x^{2}}]$

Step 4

Differentiate.

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Step 4.1

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

$x^{\frac{1}{2}} e^{x^{2}} (10 {(x^{2} + 1)}^{9} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1])) + {(x^{2} + 1)}^{10} \frac{d}{d x} [x^{\frac{1}{2}} e^{x^{2}}]$

Step 4.2

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

$x^{\frac{1}{2}} e^{x^{2}} (10 {(x^{2} + 1)}^{9} (2 x + \frac{d}{d x} [1])) + {(x^{2} + 1)}^{10} \frac{d}{d x} [x^{\frac{1}{2}} e^{x^{2}}]$

Step 4.3

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

$x^{\frac{1}{2}} e^{x^{2}} (10 {(x^{2} + 1)}^{9} (2 x + 0)) + {(x^{2} + 1)}^{10} \frac{d}{d x} [x^{\frac{1}{2}} e^{x^{2}}]$

Step 4.4

Simplify the expression.

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Step 4.4.1

Add $2 x$ and $0$ .

$x^{\frac{1}{2}} e^{x^{2}} (10 {(x^{2} + 1)}^{9} (2 x)) + {(x^{2} + 1)}^{10} \frac{d}{d x} [x^{\frac{1}{2}} e^{x^{2}}]$

Step 4.4.2

Multiply $2$ by $10$ .

$x^{\frac{1}{2}} e^{x^{2}} (20 {(x^{2} + 1)}^{9} x) + {(x^{2} + 1)}^{10} \frac{d}{d x} [x^{\frac{1}{2}} e^{x^{2}}]$

Step 5

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

$x^{1} x^{\frac{1}{2}} e^{x^{2}} (20 {(x^{2} + 1)}^{9}) + {(x^{2} + 1)}^{10} \frac{d}{d x} [x^{\frac{1}{2}} e^{x^{2}}]$

Step 6

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

$x^{1 + \frac{1}{2}} e^{x^{2}} (20 {(x^{2} + 1)}^{9}) + {(x^{2} + 1)}^{10} \frac{d}{d x} [x^{\frac{1}{2}} e^{x^{2}}]$

Step 7

Simplify the expression.

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Step 7.1

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

$x^{\frac{2}{2} + \frac{1}{2}} e^{x^{2}} (20 {(x^{2} + 1)}^{9}) + {(x^{2} + 1)}^{10} \frac{d}{d x} [x^{\frac{1}{2}} e^{x^{2}}]$

Step 7.2

Combine the numerators over the common denominator.

$x^{\frac{2 + 1}{2}} e^{x^{2}} (20 {(x^{2} + 1)}^{9}) + {(x^{2} + 1)}^{10} \frac{d}{d x} [x^{\frac{1}{2}} e^{x^{2}}]$

Step 7.3

Add $2$ and $1$ .

$x^{\frac{3}{2}} e^{x^{2}} (20 {(x^{2} + 1)}^{9}) + {(x^{2} + 1)}^{10} \frac{d}{d x} [x^{\frac{1}{2}} e^{x^{2}}]$

Step 8

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 $f (x) = x^{\frac{1}{2}}$ and $g (x) = e^{x^{2}}$ .

$x^{\frac{3}{2}} e^{x^{2}} (20 {(x^{2} + 1)}^{9}) + {(x^{2} + 1)}^{10} (x^{\frac{1}{2}} \frac{d}{d x} [e^{x^{2}}] + e^{x^{2}} \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 9

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 9.1

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

$x^{\frac{3}{2}} e^{x^{2}} (20 {(x^{2} + 1)}^{9}) + {(x^{2} + 1)}^{10} (x^{\frac{1}{2}} (\frac{d}{d u_{2}} [e^{u_{2}}] \frac{d}{d x} [x^{2}]) + e^{x^{2}} \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 9.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{2}} [a^{u_{2}}]$ is $a^{u_{2}} \ln (a)$ where $a$ = $e$ .

$x^{\frac{3}{2}} e^{x^{2}} (20 {(x^{2} + 1)}^{9}) + {(x^{2} + 1)}^{10} (x^{\frac{1}{2}} (e^{u_{2}} \frac{d}{d x} [x^{2}]) + e^{x^{2}} \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 9.3

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

$x^{\frac{3}{2}} e^{x^{2}} (20 {(x^{2} + 1)}^{9}) + {(x^{2} + 1)}^{10} (x^{\frac{1}{2}} (e^{x^{2}} \frac{d}{d x} [x^{2}]) + e^{x^{2}} \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 10

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

$x^{\frac{3}{2}} e^{x^{2}} (20 {(x^{2} + 1)}^{9}) + {(x^{2} + 1)}^{10} (x^{\frac{1}{2}} (e^{x^{2}} (2 x)) + e^{x^{2}} \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 11

Multiply $x^{\frac{1}{2}}$ by $x$ by adding the exponents.

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Step 11.1

Move $x$ .

$x^{\frac{3}{2}} e^{x^{2}} (20 {(x^{2} + 1)}^{9}) + {(x^{2} + 1)}^{10} (x \cdot x^{\frac{1}{2}} (e^{x^{2}} \cdot (2)) + e^{x^{2}} \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 11.2

Multiply $x$ by $x^{\frac{1}{2}}$ .

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Step 11.2.1

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

$x^{\frac{3}{2}} e^{x^{2}} (20 {(x^{2} + 1)}^{9}) + {(x^{2} + 1)}^{10} (x^{1} x^{\frac{1}{2}} (e^{x^{2}} \cdot (2)) + e^{x^{2}} \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 11.2.2

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

$x^{\frac{3}{2}} e^{x^{2}} (20 {(x^{2} + 1)}^{9}) + {(x^{2} + 1)}^{10} (x^{1 + \frac{1}{2}} (e^{x^{2}} \cdot (2)) + e^{x^{2}} \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 11.3

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

$x^{\frac{3}{2}} e^{x^{2}} (20 {(x^{2} + 1)}^{9}) + {(x^{2} + 1)}^{10} (x^{\frac{2}{2} + \frac{1}{2}} (e^{x^{2}} \cdot (2)) + e^{x^{2}} \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 11.4

Combine the numerators over the common denominator.

$x^{\frac{3}{2}} e^{x^{2}} (20 {(x^{2} + 1)}^{9}) + {(x^{2} + 1)}^{10} (x^{\frac{2 + 1}{2}} (e^{x^{2}} \cdot (2)) + e^{x^{2}} \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 11.5

Add $2$ and $1$ .

$x^{\frac{3}{2}} e^{x^{2}} (20 {(x^{2} + 1)}^{9}) + {(x^{2} + 1)}^{10} (x^{\frac{3}{2}} (e^{x^{2}} \cdot (2)) + e^{x^{2}} \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 12

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

$x^{\frac{3}{2}} e^{x^{2}} (20 {(x^{2} + 1)}^{9}) + {(x^{2} + 1)}^{10} (x^{\frac{3}{2}} (2 \cdot e^{x^{2}}) + e^{x^{2}} \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 13

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

$x^{\frac{3}{2}} e^{x^{2}} (20 {(x^{2} + 1)}^{9}) + {(x^{2} + 1)}^{10} (x^{\frac{3}{2}} (2 e^{x^{2}}) + e^{x^{2}} (\frac{1}{2} x^{\frac{1}{2} - 1}))$

Step 14

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

$x^{\frac{3}{2}} e^{x^{2}} (20 {(x^{2} + 1)}^{9}) + {(x^{2} + 1)}^{10} (x^{\frac{3}{2}} (2 e^{x^{2}}) + e^{x^{2}} (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}))$

Step 15

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

$x^{\frac{3}{2}} e^{x^{2}} (20 {(x^{2} + 1)}^{9}) + {(x^{2} + 1)}^{10} (x^{\frac{3}{2}} (2 e^{x^{2}}) + e^{x^{2}} (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}))$

Step 16

Combine the numerators over the common denominator.

$x^{\frac{3}{2}} e^{x^{2}} (20 {(x^{2} + 1)}^{9}) + {(x^{2} + 1)}^{10} (x^{\frac{3}{2}} (2 e^{x^{2}}) + e^{x^{2}} (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}}))$

Step 17

Simplify the numerator.

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Step 17.1

Multiply $- 1$ by $2$ .

$x^{\frac{3}{2}} e^{x^{2}} (20 {(x^{2} + 1)}^{9}) + {(x^{2} + 1)}^{10} (x^{\frac{3}{2}} (2 e^{x^{2}}) + e^{x^{2}} (\frac{1}{2} x^{\frac{1 - 2}{2}}))$

Step 17.2

Subtract $2$ from $1$ .

$x^{\frac{3}{2}} e^{x^{2}} (20 {(x^{2} + 1)}^{9}) + {(x^{2} + 1)}^{10} (x^{\frac{3}{2}} (2 e^{x^{2}}) + e^{x^{2}} (\frac{1}{2} x^{\frac{- 1}{2}}))$

Step 18

Move the negative in front of the fraction.

$x^{\frac{3}{2}} e^{x^{2}} (20 {(x^{2} + 1)}^{9}) + {(x^{2} + 1)}^{10} (x^{\frac{3}{2}} (2 e^{x^{2}}) + e^{x^{2}} (\frac{1}{2} x^{- \frac{1}{2}}))$

Step 19

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

$x^{\frac{3}{2}} e^{x^{2}} (20 {(x^{2} + 1)}^{9}) + {(x^{2} + 1)}^{10} (x^{\frac{3}{2}} (2 e^{x^{2}}) + e^{x^{2}} \frac{x^{- \frac{1}{2}}}{2})$

Step 20

Combine $e^{x^{2}}$ and $\frac{x^{- \frac{1}{2}}}{2}$ .

$x^{\frac{3}{2}} e^{x^{2}} (20 {(x^{2} + 1)}^{9}) + {(x^{2} + 1)}^{10} (x^{\frac{3}{2}} (2 e^{x^{2}}) + \frac{e^{x^{2}} x^{- \frac{1}{2}}}{2})$

Step 21

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

$x^{\frac{3}{2}} e^{x^{2}} (20 {(x^{2} + 1)}^{9}) + {(x^{2} + 1)}^{10} (x^{\frac{3}{2}} (2 e^{x^{2}}) + \frac{e^{x^{2}}}{2 x^{\frac{1}{2}}})$

Step 22

Simplify.

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Step 22.1

Reorder terms.

$20 e^{x^{2}} x^{\frac{3}{2}} {(x^{2} + 1)}^{9} + {(x^{2} + 1)}^{10} (x^{\frac{3}{2}} (2 e^{x^{2}}) + \frac{e^{x^{2}}}{2 x^{\frac{1}{2}}})$

Step 22.2

Simplify each term.

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Step 22.2.1

Rewrite using the commutative property of multiplication.

$20 e^{x^{2}} x^{\frac{3}{2}} {(x^{2} + 1)}^{9} + {(x^{2} + 1)}^{10} (2 x^{\frac{3}{2}} e^{x^{2}} + \frac{e^{x^{2}}}{2 x^{\frac{1}{2}}})$

Step 22.2.2

Apply the distributive property.

$20 e^{x^{2}} x^{\frac{3}{2}} {(x^{2} + 1)}^{9} + {(x^{2} + 1)}^{10} (2 x^{\frac{3}{2}} e^{x^{2}}) + {(x^{2} + 1)}^{10} \frac{e^{x^{2}}}{2 x^{\frac{1}{2}}}$

Step 22.2.3

Rewrite using the commutative property of multiplication.

$20 e^{x^{2}} x^{\frac{3}{2}} {(x^{2} + 1)}^{9} + 2 {(x^{2} + 1)}^{10} x^{\frac{3}{2}} e^{x^{2}} + {(x^{2} + 1)}^{10} \frac{e^{x^{2}}}{2 x^{\frac{1}{2}}}$

Step 22.2.4

Combine ${(x^{2} + 1)}^{10}$ and $\frac{e^{x^{2}}}{2 x^{\frac{1}{2}}}$ .

$20 e^{x^{2}} x^{\frac{3}{2}} {(x^{2} + 1)}^{9} + 2 {(x^{2} + 1)}^{10} x^{\frac{3}{2}} e^{x^{2}} + \frac{{(x^{2} + 1)}^{10} e^{x^{2}}}{2 x^{\frac{1}{2}}}$

Step 22.2.5

To write $2 {(x^{2} + 1)}^{10} x^{\frac{3}{2}} e^{x^{2}}$ as a fraction with a common denominator, multiply by $\frac{2 x^{\frac{1}{2}}}{2 x^{\frac{1}{2}}}$ .

$20 e^{x^{2}} x^{\frac{3}{2}} {(x^{2} + 1)}^{9} + 2 {(x^{2} + 1)}^{10} x^{\frac{3}{2}} e^{x^{2}} \cdot \frac{2 x^{\frac{1}{2}}}{2 x^{\frac{1}{2}}} + \frac{{(x^{2} + 1)}^{10} e^{x^{2}}}{2 x^{\frac{1}{2}}}$

Step 22.2.6

Combine $2 {(x^{2} + 1)}^{10} x^{\frac{3}{2}} e^{x^{2}}$ and $\frac{2 x^{\frac{1}{2}}}{2 x^{\frac{1}{2}}}$ .

$20 e^{x^{2}} x^{\frac{3}{2}} {(x^{2} + 1)}^{9} + \frac{2 {(x^{2} + 1)}^{10} x^{\frac{3}{2}} e^{x^{2}} (2 x^{\frac{1}{2}})}{2 x^{\frac{1}{2}}} + \frac{{(x^{2} + 1)}^{10} e^{x^{2}}}{2 x^{\frac{1}{2}}}$

Step 22.2.7

Combine the numerators over the common denominator.

$20 e^{x^{2}} x^{\frac{3}{2}} {(x^{2} + 1)}^{9} + \frac{2 {(x^{2} + 1)}^{10} x^{\frac{3}{2}} e^{x^{2}} (2 x^{\frac{1}{2}}) + {(x^{2} + 1)}^{10} e^{x^{2}}}{2 x^{\frac{1}{2}}}$

Step 22.2.8

Simplify the numerator.

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Step 22.2.8.1

Factor ${(x^{2} + 1)}^{10} e^{x^{2}}$ out of $2 {(x^{2} + 1)}^{10} x^{\frac{3}{2}} e^{x^{2}} (2 x^{\frac{1}{2}}) + {(x^{2} + 1)}^{10} e^{x^{2}}$ .

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Step 22.2.8.1.1

Factor ${(x^{2} + 1)}^{10} e^{x^{2}}$ out of $2 {(x^{2} + 1)}^{10} x^{\frac{3}{2}} e^{x^{2}} (2 x^{\frac{1}{2}})$ .

$20 e^{x^{2}} x^{\frac{3}{2}} {(x^{2} + 1)}^{9} + \frac{{(x^{2} + 1)}^{10} e^{x^{2}} (2 x^{\frac{3}{2}} (2 x^{\frac{1}{2}})) + {(x^{2} + 1)}^{10} e^{x^{2}}}{2 x^{\frac{1}{2}}}$

Step 22.2.8.1.2

Factor ${(x^{2} + 1)}^{10} e^{x^{2}}$ out of ${(x^{2} + 1)}^{10} e^{x^{2}}$ .

$20 e^{x^{2}} x^{\frac{3}{2}} {(x^{2} + 1)}^{9} + \frac{{(x^{2} + 1)}^{10} e^{x^{2}} (2 x^{\frac{3}{2}} (2 x^{\frac{1}{2}})) + {(x^{2} + 1)}^{10} e^{x^{2}} (1)}{2 x^{\frac{1}{2}}}$

Step 22.2.8.1.3

Factor ${(x^{2} + 1)}^{10} e^{x^{2}}$ out of ${(x^{2} + 1)}^{10} e^{x^{2}} (2 x^{\frac{3}{2}} (2 x^{\frac{1}{2}})) + {(x^{2} + 1)}^{10} e^{x^{2}} (1)$ .

$20 e^{x^{2}} x^{\frac{3}{2}} {(x^{2} + 1)}^{9} + \frac{{(x^{2} + 1)}^{10} e^{x^{2}} (2 x^{\frac{3}{2}} (2 x^{\frac{1}{2}}) + 1)}{2 x^{\frac{1}{2}}}$

Step 22.2.8.2

Combine exponents.

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Step 22.2.8.2.1

Multiply $2$ by $2$ .

$20 e^{x^{2}} x^{\frac{3}{2}} {(x^{2} + 1)}^{9} + \frac{{(x^{2} + 1)}^{10} e^{x^{2}} (4 x^{\frac{3}{2}} x^{\frac{1}{2}} + 1)}{2 x^{\frac{1}{2}}}$

Step 22.2.8.2.2

Multiply $x^{\frac{3}{2}}$ by $x^{\frac{1}{2}}$ by adding the exponents.

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Step 22.2.8.2.2.1

Move $x^{\frac{1}{2}}$ .

$20 e^{x^{2}} x^{\frac{3}{2}} {(x^{2} + 1)}^{9} + \frac{{(x^{2} + 1)}^{10} e^{x^{2}} (4 (x^{\frac{1}{2}} x^{\frac{3}{2}}) + 1)}{2 x^{\frac{1}{2}}}$

Step 22.2.8.2.2.2

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

$20 e^{x^{2}} x^{\frac{3}{2}} {(x^{2} + 1)}^{9} + \frac{{(x^{2} + 1)}^{10} e^{x^{2}} (4 x^{\frac{1}{2} + \frac{3}{2}} + 1)}{2 x^{\frac{1}{2}}}$

Step 22.2.8.2.2.3

Combine the numerators over the common denominator.

$20 e^{x^{2}} x^{\frac{3}{2}} {(x^{2} + 1)}^{9} + \frac{{(x^{2} + 1)}^{10} e^{x^{2}} (4 x^{\frac{1 + 3}{2}} + 1)}{2 x^{\frac{1}{2}}}$

Step 22.2.8.2.2.4

Add $1$ and $3$ .

$20 e^{x^{2}} x^{\frac{3}{2}} {(x^{2} + 1)}^{9} + \frac{{(x^{2} + 1)}^{10} e^{x^{2}} (4 x^{\frac{4}{2}} + 1)}{2 x^{\frac{1}{2}}}$

Step 22.2.8.2.2.5

Divide $4$ by $2$ .

$20 e^{x^{2}} x^{\frac{3}{2}} {(x^{2} + 1)}^{9} + \frac{{(x^{2} + 1)}^{10} e^{x^{2}} (4 x^{2} + 1)}{2 x^{\frac{1}{2}}}$

Step 22.3

To write $20 e^{x^{2}} x^{\frac{3}{2}} {(x^{2} + 1)}^{9}$ as a fraction with a common denominator, multiply by $\frac{2 x^{\frac{1}{2}}}{2 x^{\frac{1}{2}}}$ .

$20 e^{x^{2}} x^{\frac{3}{2}} {(x^{2} + 1)}^{9} \cdot \frac{2 x^{\frac{1}{2}}}{2 x^{\frac{1}{2}}} + \frac{{(x^{2} + 1)}^{10} e^{x^{2}} (4 x^{2} + 1)}{2 x^{\frac{1}{2}}}$

Step 22.4

Combine $20 e^{x^{2}} x^{\frac{3}{2}} {(x^{2} + 1)}^{9}$ and $\frac{2 x^{\frac{1}{2}}}{2 x^{\frac{1}{2}}}$ .

$\frac{20 e^{x^{2}} x^{\frac{3}{2}} {(x^{2} + 1)}^{9} (2 x^{\frac{1}{2}})}{2 x^{\frac{1}{2}}} + \frac{{(x^{2} + 1)}^{10} e^{x^{2}} (4 x^{2} + 1)}{2 x^{\frac{1}{2}}}$

Step 22.5

Combine the numerators over the common denominator.

$\frac{20 e^{x^{2}} x^{\frac{3}{2}} {(x^{2} + 1)}^{9} (2 x^{\frac{1}{2}}) + {(x^{2} + 1)}^{10} e^{x^{2}} (4 x^{2} + 1)}{2 x^{\frac{1}{2}}}$

Step 22.6

Simplify the numerator.

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Step 22.6.1

Factor ${(x^{2} + 1)}^{9}$ out of $20 e^{x^{2}} x^{\frac{3}{2}} {(x^{2} + 1)}^{9} (2 x^{\frac{1}{2}}) + {(x^{2} + 1)}^{10} e^{x^{2}} (4 x^{2} + 1)$ .

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Step 22.6.1.1

Factor ${(x^{2} + 1)}^{9}$ out of $20 e^{x^{2}} x^{\frac{3}{2}} {(x^{2} + 1)}^{9} (2 x^{\frac{1}{2}})$ .

$\frac{{(x^{2} + 1)}^{9} (20 e^{x^{2}} x^{\frac{3}{2}} (2 x^{\frac{1}{2}})) + {(x^{2} + 1)}^{10} e^{x^{2}} (4 x^{2} + 1)}{2 x^{\frac{1}{2}}}$

Step 22.6.1.2

Factor ${(x^{2} + 1)}^{9}$ out of ${(x^{2} + 1)}^{10} e^{x^{2}} (4 x^{2} + 1)$ .

$\frac{{(x^{2} + 1)}^{9} (20 e^{x^{2}} x^{\frac{3}{2}} (2 x^{\frac{1}{2}})) + {(x^{2} + 1)}^{9} ((x^{2} + 1) e^{x^{2}} (4 x^{2} + 1))}{2 x^{\frac{1}{2}}}$

Step 22.6.1.3

Factor ${(x^{2} + 1)}^{9}$ out of ${(x^{2} + 1)}^{9} (20 e^{x^{2}} x^{\frac{3}{2}} (2 x^{\frac{1}{2}})) + {(x^{2} + 1)}^{9} ((x^{2} + 1) e^{x^{2}} (4 x^{2} + 1))$ .

$\frac{{(x^{2} + 1)}^{9} (20 e^{x^{2}} x^{\frac{3}{2}} (2 x^{\frac{1}{2}}) + (x^{2} + 1) e^{x^{2}} (4 x^{2} + 1))}{2 x^{\frac{1}{2}}}$

Step 22.6.2

Factor $e^{x^{2}}$ out of $20 e^{x^{2}} x^{\frac{3}{2}} (2 x^{\frac{1}{2}}) + (x^{2} + 1) e^{x^{2}} (4 x^{2} + 1)$ .

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Step 22.6.2.1

Factor $e^{x^{2}}$ out of $20 e^{x^{2}} x^{\frac{3}{2}} (2 x^{\frac{1}{2}})$ .

$\frac{{(x^{2} + 1)}^{9} (e^{x^{2}} (20 x^{\frac{3}{2}} (2 x^{\frac{1}{2}})) + (x^{2} + 1) e^{x^{2}} (4 x^{2} + 1))}{2 x^{\frac{1}{2}}}$

Step 22.6.2.2

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

$\frac{{(x^{2} + 1)}^{9} (e^{x^{2}} (20 x^{\frac{3}{2}} (2 x^{\frac{1}{2}})) + e^{x^{2}} ((x^{2} + 1) (4 x^{2} + 1)))}{2 x^{\frac{1}{2}}}$

Step 22.6.2.3

Factor $e^{x^{2}}$ out of $e^{x^{2}} (20 x^{\frac{3}{2}} (2 x^{\frac{1}{2}})) + e^{x^{2}} ((x^{2} + 1) (4 x^{2} + 1))$ .

$\frac{{(x^{2} + 1)}^{9} (e^{x^{2}} (20 x^{\frac{3}{2}} (2 x^{\frac{1}{2}}) + (x^{2} + 1) (4 x^{2} + 1)))}{2 x^{\frac{1}{2}}}$

Step 22.6.3

Rewrite using the commutative property of multiplication.

$\frac{{(x^{2} + 1)}^{9} (e^{x^{2}} (20 \cdot 2 x^{\frac{3}{2}} x^{\frac{1}{2}} + (x^{2} + 1) (4 x^{2} + 1)))}{2 x^{\frac{1}{2}}}$

Step 22.6.4

Multiply $x^{\frac{3}{2}}$ by $x^{\frac{1}{2}}$ by adding the exponents.

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Step 22.6.4.1

Move $x^{\frac{1}{2}}$ .

$\frac{{(x^{2} + 1)}^{9} (e^{x^{2}} (20 \cdot 2 (x^{\frac{1}{2}} x^{\frac{3}{2}}) + (x^{2} + 1) (4 x^{2} + 1)))}{2 x^{\frac{1}{2}}}$

Step 22.6.4.2

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

$\frac{{(x^{2} + 1)}^{9} (e^{x^{2}} (20 \cdot 2 x^{\frac{1}{2} + \frac{3}{2}} + (x^{2} + 1) (4 x^{2} + 1)))}{2 x^{\frac{1}{2}}}$

Step 22.6.4.3

Combine the numerators over the common denominator.

$\frac{{(x^{2} + 1)}^{9} (e^{x^{2}} (20 \cdot 2 x^{\frac{1 + 3}{2}} + (x^{2} + 1) (4 x^{2} + 1)))}{2 x^{\frac{1}{2}}}$

Step 22.6.4.4

Add $1$ and $3$ .

$\frac{{(x^{2} + 1)}^{9} (e^{x^{2}} (20 \cdot 2 x^{\frac{4}{2}} + (x^{2} + 1) (4 x^{2} + 1)))}{2 x^{\frac{1}{2}}}$

Step 22.6.4.5

Divide $4$ by $2$ .

$\frac{{(x^{2} + 1)}^{9} (e^{x^{2}} (20 \cdot 2 x^{2} + (x^{2} + 1) (4 x^{2} + 1)))}{2 x^{\frac{1}{2}}}$

Step 22.6.5

Multiply $20$ by $2$ .

$\frac{{(x^{2} + 1)}^{9} (e^{x^{2}} (40 x^{2} + (x^{2} + 1) (4 x^{2} + 1)))}{2 x^{\frac{1}{2}}}$

Step 22.6.6

Expand $(x^{2} + 1) (4 x^{2} + 1)$ using the FOIL Method.

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Step 22.6.6.1

Apply the distributive property.

$\frac{{(x^{2} + 1)}^{9} (e^{x^{2}} (40 x^{2} + x^{2} (4 x^{2} + 1) + 1 (4 x^{2} + 1)))}{2 x^{\frac{1}{2}}}$

Step 22.6.6.2

Apply the distributive property.

$\frac{{(x^{2} + 1)}^{9} (e^{x^{2}} (40 x^{2} + x^{2} (4 x^{2}) + x^{2} \cdot 1 + 1 (4 x^{2} + 1)))}{2 x^{\frac{1}{2}}}$

Step 22.6.6.3

Apply the distributive property.

$\frac{{(x^{2} + 1)}^{9} (e^{x^{2}} (40 x^{2} + x^{2} (4 x^{2}) + x^{2} \cdot 1 + 1 (4 x^{2}) + 1 \cdot 1))}{2 x^{\frac{1}{2}}}$

Step 22.6.7

Simplify and combine like terms.

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Step 22.6.7.1

Simplify each term.

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Step 22.6.7.1.1

Rewrite using the commutative property of multiplication.

$\frac{{(x^{2} + 1)}^{9} (e^{x^{2}} (40 x^{2} + 4 x^{2} x^{2} + x^{2} \cdot 1 + 1 (4 x^{2}) + 1 \cdot 1))}{2 x^{\frac{1}{2}}}$

Step 22.6.7.1.2

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

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Step 22.6.7.1.2.1

Move $x^{2}$ .

$\frac{{(x^{2} + 1)}^{9} (e^{x^{2}} (40 x^{2} + 4 (x^{2} x^{2}) + x^{2} \cdot 1 + 1 (4 x^{2}) + 1 \cdot 1))}{2 x^{\frac{1}{2}}}$

Step 22.6.7.1.2.2

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

$\frac{{(x^{2} + 1)}^{9} (e^{x^{2}} (40 x^{2} + 4 x^{2 + 2} + x^{2} \cdot 1 + 1 (4 x^{2}) + 1 \cdot 1))}{2 x^{\frac{1}{2}}}$

Step 22.6.7.1.2.3

Add $2$ and $2$ .

$\frac{{(x^{2} + 1)}^{9} (e^{x^{2}} (40 x^{2} + 4 x^{4} + x^{2} \cdot 1 + 1 (4 x^{2}) + 1 \cdot 1))}{2 x^{\frac{1}{2}}}$

Step 22.6.7.1.3

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

$\frac{{(x^{2} + 1)}^{9} (e^{x^{2}} (40 x^{2} + 4 x^{4} + x^{2} + 1 (4 x^{2}) + 1 \cdot 1))}{2 x^{\frac{1}{2}}}$

Step 22.6.7.1.4

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

$\frac{{(x^{2} + 1)}^{9} (e^{x^{2}} (40 x^{2} + 4 x^{4} + x^{2} + 4 x^{2} + 1 \cdot 1))}{2 x^{\frac{1}{2}}}$

Step 22.6.7.1.5

Multiply $1$ by $1$ .

$\frac{{(x^{2} + 1)}^{9} (e^{x^{2}} (40 x^{2} + 4 x^{4} + x^{2} + 4 x^{2} + 1))}{2 x^{\frac{1}{2}}}$

Step 22.6.7.2

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

$\frac{{(x^{2} + 1)}^{9} (e^{x^{2}} (40 x^{2} + 4 x^{4} + 5 x^{2} + 1))}{2 x^{\frac{1}{2}}}$

Step 22.6.8

Add $40 x^{2}$ and $5 x^{2}$ .

$\frac{{(x^{2} + 1)}^{9} e^{x^{2}} (4 x^{4} + 45 x^{2} + 1)}{2 x^{\frac{1}{2}}}$

Step 22.7

Reorder factors in $\frac{{(x^{2} + 1)}^{9} e^{x^{2}} (4 x^{4} + 45 x^{2} + 1)}{2 x^{\frac{1}{2}}}$ .

$\frac{e^{x^{2}} {(x^{2} + 1)}^{9} (4 x^{4} + 45 x^{2} + 1)}{2 x^{\frac{1}{2}}}$