Calculus Examples

$f (x) = e^{\cos^{5} (x)}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = \cos^{5} (x)$ .

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as $\cos^{5} (x)$ .

$\frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d x} [\cos^{5} (x)]$

Step 1.2

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

$e^{u_{1}} \frac{d}{d x} [\cos^{5} (x)]$

Step 1.3

Replace all occurrences of $u_{1}$ with $\cos^{5} (x)$ .

$e^{\cos^{5} (x)} \frac{d}{d x} [\cos^{5} (x)]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{5}$ and $g (x) = \cos (x)$ .

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Step 2.1

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

$e^{\cos^{5} (x)} (\frac{d}{d u_{2}} [{u_{2}}^{5}] \frac{d}{d x} [\cos (x)])$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 5$ .

$e^{\cos^{5} (x)} (5 {u_{2}}^{4} \frac{d}{d x} [\cos (x)])$

Step 2.3

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

$e^{\cos^{5} (x)} (5 \cos^{4} (x) \frac{d}{d x} [\cos (x)])$

Step 3

Move $5$ to the left of $e^{\cos^{5} (x)}$ .

$5 \cdot e^{\cos^{5} (x)} \cos^{4} (x) \frac{d}{d x} [\cos (x)]$

Step 4

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

$5 e^{\cos^{5} (x)} \cos^{4} (x) (- \sin (x))$

Step 5

Multiply $- 1$ by $5$ .

$- 5 e^{\cos^{5} (x)} \cos^{4} (x) \sin (x)$