Calculus Examples

$y = \cos (\cos (x))$

Step 1

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

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Step 1.1

To apply the Chain Rule, set $u$ as $\cos (x)$ .

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

Step 1.2

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

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

Step 1.3

Replace all occurrences of $u$ with $\cos (x)$ .

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

Step 2

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

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

Step 3

Simplify the expression.

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Step 3.1

Multiply $- 1$ by $- 1$ .

$1 \sin (\cos (x)) \sin (x)$

Step 3.2

Multiply $\sin (\cos (x))$ by $1$ .

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

Step 3.3

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

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