Calculus Examples

$f (x) = 2 x^{2} + 7 x - \ln (x^{2} + 1)$

Step 1

By the Sum Rule, the derivative of $2 x^{2} + 7 x - \ln (x^{2} + 1)$ with respect to $x$ is $\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [7 x] + \frac{d}{d x} [- \ln (x^{2} + 1)]$ .

$\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [7 x] + \frac{d}{d x} [- \ln (x^{2} + 1)]$

Step 2

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

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Step 2.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{2}$ with respect to $x$ is $2 \frac{d}{d x} [x^{2}]$ .

$2 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [7 x] + \frac{d}{d x} [- \ln (x^{2} + 1)]$

Step 2.2

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

$2 (2 x) + \frac{d}{d x} [7 x] + \frac{d}{d x} [- \ln (x^{2} + 1)]$

Step 2.3

Multiply $2$ by $2$ .

$4 x + \frac{d}{d x} [7 x] + \frac{d}{d x} [- \ln (x^{2} + 1)]$

Step 3

Evaluate $\frac{d}{d x} [7 x]$ .

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Step 3.1

Since $7$ is constant with respect to $x$ , the derivative of $7 x$ with respect to $x$ is $7 \frac{d}{d x} [x]$ .

$4 x + 7 \frac{d}{d x} [x] + \frac{d}{d x} [- \ln (x^{2} + 1)]$

Step 3.2

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

$4 x + 7 \cdot 1 + \frac{d}{d x} [- \ln (x^{2} + 1)]$

Step 3.3

Multiply $7$ by $1$ .

$4 x + 7 + \frac{d}{d x} [- \ln (x^{2} + 1)]$

Step 4

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

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Step 4.1

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

$4 x + 7 - \frac{d}{d x} [\ln (x^{2} + 1)]$

Step 4.2

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 4.2.1

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

$4 x + 7 - (\frac{d}{d u} [\ln (u)] \frac{d}{d x} [x^{2} + 1])$

Step 4.2.2

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

$4 x + 7 - (\frac{1}{u} \frac{d}{d x} [x^{2} + 1])$

Step 4.2.3

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

$4 x + 7 - (\frac{1}{x^{2} + 1} \frac{d}{d x} [x^{2} + 1])$

Step 4.3

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

$4 x + 7 - (\frac{1}{x^{2} + 1} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]))$

Step 4.4

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

$4 x + 7 - (\frac{1}{x^{2} + 1} (2 x + \frac{d}{d x} [1]))$

Step 4.5

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

$4 x + 7 - (\frac{1}{x^{2} + 1} (2 x + 0))$

Step 4.6

Add $2 x$ and $0$ .

$4 x + 7 - (\frac{1}{x^{2} + 1} (2 x))$

Step 4.7

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

$4 x + 7 - (\frac{2}{x^{2} + 1} x)$

Step 4.8

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

$4 x + 7 - \frac{2 x}{x^{2} + 1}$

Step 5

Combine terms.

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Step 5.1

To write $4 x$ as a fraction with a common denominator, multiply by $\frac{x^{2} + 1}{x^{2} + 1}$ .

$\frac{4 x (x^{2} + 1)}{x^{2} + 1} - \frac{2 x}{x^{2} + 1} + 7$

Step 5.2

Combine the numerators over the common denominator.

$\frac{4 x (x^{2} + 1) - 2 x}{x^{2} + 1} + 7$

Step 5.3

To write $7$ as a fraction with a common denominator, multiply by $\frac{x^{2} + 1}{x^{2} + 1}$ .

$\frac{4 x (x^{2} + 1) - 2 x}{x^{2} + 1} + \frac{7 (x^{2} + 1)}{x^{2} + 1}$

Step 5.4

Combine the numerators over the common denominator.

$\frac{4 x (x^{2} + 1) - 2 x + 7 (x^{2} + 1)}{x^{2} + 1}$