Calculus Examples

sin (x^{2})

$\sin (x^{2})$

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

$f (x) = \sin (x)$ and

$g (x) = x^{2}$ .

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Step 1.1

To apply the Chain Rule, set

$u$ as

$x^{2}$ .

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

Step 1.2

The derivative of

$\sin (u)$ with respect to

$u$ is

$\cos (u)$ .

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

Step 1.3

Replace all occurrences of

$u$ with

$x^{2}$ .

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

Step 2

Differentiate using the Power Rule.

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Step 2.1

Differentiate using the Power Rule which states that

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

$n x^{n - 1}$ where

$n = 2$ .

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

Step 2.2

Reorder the factors of

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

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