Calculus Examples

$e^{\sqrt{\cos (x)}}$

Step 1

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $e$ .

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

Step 2.3

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

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

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 $g (x) = \cos (x)$ .

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Step 3.1

To apply the Chain Rule, set $u_{2}$ as $\cos (x)$ .

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

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 $n = \frac{1}{2}$ .

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

Step 3.3

Replace all occurrences of $u_{2}$ with $\cos (x)$ .

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

Step 4

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

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

Step 5

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

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $2$ .

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

Step 7.2

Subtract $2$ from $1$ .

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

Step 8

Combine fractions.

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Step 8.1

Move the negative in front of the fraction.

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

Step 8.2

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

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

Step 8.3

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

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

Step 8.4

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

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

Step 9

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

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

Step 10

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

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