Calculus Examples

$y = \arctan (x^{2})$

Step 1

Differentiate using the chain rule, which states that

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

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

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

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

Tap for more steps...

Step 1.1

To apply the Chain Rule, set

$u$ as

$x^{2}$ .

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

Step 1.2

The derivative of

$\arctan (u)$ with respect to

$u$ is

$\frac{1}{1 + u^{2}}$ .

$\frac{1}{1 + u^{2}} \frac{d}{d x} [x^{2}]$

Step 1.3

Replace all occurrences of

$u$ with

$x^{2}$ .

$\frac{1}{1 + {(x^{2})}^{2}} \frac{d}{d x} [x^{2}]$

Step 2

Differentiate using the Power Rule.

Tap for more steps...

Step 2.1

Multiply the exponents in

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

Tap for more steps...

Step 2.1.1

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{1 + x^{2 \cdot 2}} \frac{d}{d x} [x^{2}]$

Step 2.1.2

Multiply

$2$ by

$2$ .

$\frac{1}{1 + x^{4}} \frac{d}{d x} [x^{2}]$

Step 2.2

Differentiate using the Power Rule which states that

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

$n x^{n - 1}$ where

$n = 2$ .

$\frac{1}{1 + x^{4}} (2 x)$

Step 2.3

Combine fractions.

Tap for more steps...

Step 2.3.1

Combine

$2$ and

$\frac{1}{1 + x^{4}}$ .

$\frac{2}{1 + x^{4}} x$

Step 2.3.2

Combine

$\frac{2}{1 + x^{4}}$ and

$x$ .

$\frac{2 x}{1 + x^{4}}$

Step 2.3.3

Reorder terms.

$\frac{2 x}{x^{4} + 1}$