Calculus Examples

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

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

Tap for more steps...

Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

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

Tap for more steps...

Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

Multiply.

Tap for more steps...

Step 3.1

Multiply $- 1$ by $- 1$ .

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

Step 3.2

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

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

Step 4

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

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

Step 5

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

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