Calculus Examples

$f (x) = \sin (\sqrt{e^{x} + 1})$

Step 1

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $2$ .

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

Step 7.2

Subtract $2$ from $1$ .

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

Step 8

Differentiate using the Sum Rule.

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Step 8.1

Move the negative in front of the fraction.

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

Step 8.2

Combine fractions.

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Step 8.2.1

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

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

Step 8.2.2

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

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

Step 8.2.3

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

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

Step 8.3

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

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

Step 9

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

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

Step 10

Differentiate using the Constant Rule.

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Step 10.1

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

$\frac{\cos ({(e^{x} + 1)}^{\frac{1}{2}})}{2 {(e^{x} + 1)}^{\frac{1}{2}}} (e^{x} + 0)$

Step 10.2

Combine fractions.

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Step 10.2.1

Add $e^{x}$ and $0$ .

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

Step 10.2.2

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

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

Step 10.2.3

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

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