Calculus Examples

$x - 6 \ln (x^{2} + 1)$

Step 1

Differentiate.

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Step 1.1

By the Sum Rule, the derivative of $x - 6 \ln (x^{2} + 1)$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 6 \ln (x^{2} + 1)]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [- 6 \ln (x^{2} + 1)]$

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$1 + \frac{d}{d x} [- 6 \ln (x^{2} + 1)]$

Step 2

Evaluate $\frac{d}{d x} [- 6 \ln (x^{2} + 1)]$ .

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Step 2.1

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 \ln (x^{2} + 1)$ with respect to $x$ is $- 6 \frac{d}{d x} [\ln (x^{2} + 1)]$ .

$1 - 6 \frac{d}{d x} [\ln (x^{2} + 1)]$

Step 2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = x^{2} + 1$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $x^{2} + 1$ .

$1 - 6 (\frac{d}{d u} [\ln (u)] \frac{d}{d x} [x^{2} + 1])$

Step 2.2.2

The derivative of $\ln (u)$ with respect to $u$ is $\frac{1}{u}$ .

$1 - 6 (\frac{1}{u} \frac{d}{d x} [x^{2} + 1])$

Step 2.2.3

Replace all occurrences of $u$ with $x^{2} + 1$ .

$1 - 6 (\frac{1}{x^{2} + 1} \frac{d}{d x} [x^{2} + 1])$

Step 2.3

By the Sum Rule, the derivative of $x^{2} + 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]$ .

$1 - 6 (\frac{1}{x^{2} + 1} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]))$

Step 2.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$1 - 6 (\frac{1}{x^{2} + 1} (2 x + \frac{d}{d x} [1]))$

Step 2.5

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$1 - 6 (\frac{1}{x^{2} + 1} (2 x + 0))$

Step 2.6

Add $2 x$ and $0$ .

$1 - 6 (\frac{1}{x^{2} + 1} (2 x))$

Step 2.7

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

$1 - 6 (\frac{2}{x^{2} + 1} x)$

Step 2.8

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

$1 - 6 \frac{2 x}{x^{2} + 1}$

Step 2.9

Combine $- 6$ and $\frac{2 x}{x^{2} + 1}$ .

$1 + \frac{- 6 (2 x)}{x^{2} + 1}$

Step 2.10

Multiply $2$ by $- 6$ .

$1 + \frac{- 12 x}{x^{2} + 1}$

Step 2.11

Move the negative in front of the fraction.

$1 - \frac{12 x}{x^{2} + 1}$

Step 3

Simplify.

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Step 3.1

Combine terms.

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Step 3.1.1

Write $1$ as a fraction with a common denominator.

$\frac{x^{2} + 1}{x^{2} + 1} - \frac{12 x}{x^{2} + 1}$

Step 3.1.2

Combine the numerators over the common denominator.

$\frac{x^{2} + 1 - 12 x}{x^{2} + 1}$

Step 3.2

Reorder terms.

$\frac{x^{2} - 12 x + 1}{x^{2} + 1}$