Calculus Examples

$\sqrt{e^{2 x} + 2 x}$

Step 1

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

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

Step 2

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 $g (x) = e^{2 x} + 2 x$ .

Tap for more steps...

Step 2.1

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

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

Step 2.2

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

$\frac{1}{2} {u_{1}}^{\frac{1}{2} - 1} \frac{d}{d x} [e^{2 x} + 2 x]$

Step 2.3

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

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

Step 3

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

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

Step 4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

Tap for more steps...

Step 6.1

Multiply $- 1$ by $2$ .

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

Step 6.2

Subtract $2$ from $1$ .

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

Step 7

Differentiate using the Sum Rule.

Tap for more steps...

Step 7.1

Move the negative in front of the fraction.

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

Step 7.2

Combine fractions.

Tap for more steps...

Step 7.2.1

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

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

Step 7.2.2

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

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

Step 7.3

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

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

Step 8

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

Tap for more steps...

Step 8.1

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

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

Step 8.2

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

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

Step 8.3

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

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

Step 9

Differentiate.

Tap for more steps...

Step 9.1

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

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

Step 9.2

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

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

Step 9.3

Simplify the expression.

Tap for more steps...

Step 9.3.1

Multiply $2$ by $1$ .

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

Step 9.3.2

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

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

Step 9.4

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

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

Step 9.5

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

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

Step 9.6

Multiply $2$ by $1$ .

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

Step 10

Simplify.

Tap for more steps...

Step 10.1

Reorder the factors of $\frac{1}{2 {(e^{2 x} + 2 x)}^{\frac{1}{2}}} (2 e^{2 x} + 2)$ .

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

Step 10.2

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

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

Step 10.3

Factor $2$ out of $2 e^{2 x}$ .

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

Step 10.4

Factor $2$ out of $2$ .

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

Step 10.5

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

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

Step 10.6

Cancel the common factors.

Tap for more steps...

Step 10.6.1

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

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

Step 10.6.2

Cancel the common factor.

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

Step 10.6.3

Rewrite the expression.

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