Calculus Examples

\ln (x^{2} + 1)

$\ln (x^{2} + 1)$

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)

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

f (x) = \ln (x)

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

g (x) = x^{2} + 1

$g (x) = x^{2} + 1$ .

Tap for more steps...

Step 1.1

To apply the Chain Rule, set

u

$u$ as

x^{2} + 1

$x^{2} + 1$ .

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

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

Step 1.2

The derivative of

\ln (u)

$\ln (u)$ with respect to

u

$u$ is

\frac{1}{u}

$\frac{1}{u}$ .

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

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

Step 1.3

Replace all occurrences of

u

$u$ with

x^{2} + 1

$x^{2} + 1$ .

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

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

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

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

Step 2

Differentiate.

Tap for more steps...

Step 2.1

By the Sum Rule, the derivative of

x^{2} + 1

$x^{2} + 1$ with respect to

x

$x$ is

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

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

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

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

Step 2.2

Differentiate using the Power Rule which states that

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

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

n x^{n - 1}

$n x^{n - 1}$ where

n = 2

$n = 2$ .

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

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

Step 2.3

Since

1

$1$ is constant with respect to

x

$x$ , the derivative of

1

$1$ with respect to

x

$x$ is

0

$0$ .

\frac{1}{x^{2} + 1} (2 x + 0)

$\frac{1}{x^{2} + 1} (2 x + 0)$

Step 2.4

Combine fractions.

Tap for more steps...

Step 2.4.1

Add

2 x

$2 x$ and

0

$0$ .

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

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

Step 2.4.2

Combine

2

$2$ and

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

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

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

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

Step 2.4.3

Combine

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

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

x

$x$ .

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

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

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

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

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

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