Calculus Examples

$y = \sin (\sin (\sin (x)))$

Step 1

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

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

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

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

Step 4

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

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