Calculus Examples

$\cos^{2} (\cos (x))$

Step 1

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

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

The derivative of $\cos (u_{2})$ with respect to $u_{2}$ is $- \sin (u_{2})$ .

$2 \cos (\cos (x)) (- \sin (u_{2}) \frac{d}{d x} [\cos (x)])$

Step 2.3

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

$2 \cos (\cos (x)) (- \sin (\cos (x)) \frac{d}{d x} [\cos (x)])$

Step 3

Multiply $- 1$ by $2$ .

$- 2 \cos (\cos (x)) (\sin (\cos (x)) \frac{d}{d x} [\cos (x)])$

Step 4

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

$- 2 \cos (\cos (x)) \sin (\cos (x)) (- \sin (x))$

Step 5

Multiply $- 1$ by $- 2$ .

$2 \cos (\cos (x)) \sin (\cos (x)) \sin (x)$

Step 6

Simplify.

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Step 6.1

Reorder the factors of $2 \cos (\cos (x)) \sin (\cos (x)) \sin (x)$ .

$2 \cos (\cos (x)) \sin (x) \sin (\cos (x))$

Step 6.2

Reorder $2 \cos (\cos (x)) \sin (x)$ and $\sin (\cos (x))$ .

$\sin (\cos (x)) (2 \cos (\cos (x)) \sin (x))$

Step 6.3

Move parentheses.

$\sin (\cos (x)) (2 \cos (\cos (x))) \sin (x)$

Step 6.4

Reorder $\sin (\cos (x))$ and $2$ .

$2 \cdot \sin (\cos (x)) \cos (\cos (x)) \sin (x)$

Step 6.5

Apply the sine double-angle identity.

$\sin (2 \cos (x)) \sin (x)$