Calculus Examples

{sin}^{3} (x)

$\sin^{3} (x)$

Step 1

Differentiate using the chain rule, which states that

\frac{d}{d x} [f (g (x))]

$\frac{d}{d x} [f (g (x))]$ is

$f' (g (x)) g' (x)$ where

$f (x) = x^{3}$ and

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

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Step 1.1

To apply the Chain Rule, set

$u$ as

$\sin (x)$ .

$\frac{d}{d u} [u^{3}] \frac{d}{d x} [\sin (x)]$

Step 1.2

Differentiate using the Power Rule which states that

$\frac{d}{d u} [u^{n}]$ is

$n u^{n - 1}$ where

$n = 3$ .

$3 u^{2} \frac{d}{d x} [\sin (x)]$

Step 1.3

Replace all occurrences of

$u$ with

$\sin (x)$ .

$3 \sin^{2} (x) \frac{d}{d x} [\sin (x)]$

Step 2

The derivative of

$\sin (x)$ with respect to

$x$ is

$\cos (x)$ .

$3 \sin^{2} (x) \cos (x)$

$\sin^{3} x$

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