Calculus Examples

$f (x) = 66 x + \ln (3 x^{4} + 13)$

Step 1

By the Sum Rule, the derivative of $66 x + \ln (3 x^{4} + 13)$ with respect to $x$ is $\frac{d}{d x} [66 x] + \frac{d}{d x} [\ln (3 x^{4} + 13)]$ .

$\frac{d}{d x} [66 x] + \frac{d}{d x} [\ln (3 x^{4} + 13)]$

Step 2

Evaluate $\frac{d}{d x} [66 x]$ .

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Step 2.1

Since $66$ is constant with respect to $x$ , the derivative of $66 x$ with respect to $x$ is $66 \frac{d}{d x} [x]$ .

$66 \frac{d}{d x} [x] + \frac{d}{d x} [\ln (3 x^{4} + 13)]$

Step 2.2

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

$66 \cdot 1 + \frac{d}{d x} [\ln (3 x^{4} + 13)]$

Step 2.3

Multiply $66$ by $1$ .

$66 + \frac{d}{d x} [\ln (3 x^{4} + 13)]$

Step 3

Evaluate $\frac{d}{d x} [\ln (3 x^{4} + 13)]$ .

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Step 3.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = 3 x^{4} + 13$ .

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Step 3.1.1

To apply the Chain Rule, set $u$ as $3 x^{4} + 13$ .

$66 + \frac{d}{d u} [\ln (u)] \frac{d}{d x} [3 x^{4} + 13]$

Step 3.1.2

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

$66 + \frac{1}{u} \frac{d}{d x} [3 x^{4} + 13]$

Step 3.1.3

Replace all occurrences of $u$ with $3 x^{4} + 13$ .

$66 + \frac{1}{3 x^{4} + 13} \frac{d}{d x} [3 x^{4} + 13]$

Step 3.2

By the Sum Rule, the derivative of $3 x^{4} + 13$ with respect to $x$ is $\frac{d}{d x} [3 x^{4}] + \frac{d}{d x} [13]$ .

$66 + \frac{1}{3 x^{4} + 13} (\frac{d}{d x} [3 x^{4}] + \frac{d}{d x} [13])$

Step 3.3

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{4}$ with respect to $x$ is $3 \frac{d}{d x} [x^{4}]$ .

$66 + \frac{1}{3 x^{4} + 13} (3 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [13])$

Step 3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$66 + \frac{1}{3 x^{4} + 13} (3 (4 x^{3}) + \frac{d}{d x} [13])$

Step 3.5

Since $13$ is constant with respect to $x$ , the derivative of $13$ with respect to $x$ is $0$ .

$66 + \frac{1}{3 x^{4} + 13} (3 (4 x^{3}) + 0)$

Step 3.6

Multiply $4$ by $3$ .

$66 + \frac{1}{3 x^{4} + 13} (12 x^{3} + 0)$

Step 3.7

Add $12 x^{3}$ and $0$ .

$66 + \frac{1}{3 x^{4} + 13} (12 x^{3})$

Step 3.8

Combine $12$ and $\frac{1}{3 x^{4} + 13}$ .

$66 + \frac{12}{3 x^{4} + 13} x^{3}$

Step 3.9

Combine $\frac{12}{3 x^{4} + 13}$ and $x^{3}$ .

$66 + \frac{12 x^{3}}{3 x^{4} + 13}$

Step 4

Simplify.

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Step 4.1

Combine terms.

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Step 4.1.1

To write $66$ as a fraction with a common denominator, multiply by $\frac{3 x^{4} + 13}{3 x^{4} + 13}$ .

$\frac{66 (3 x^{4} + 13)}{3 x^{4} + 13} + \frac{12 x^{3}}{3 x^{4} + 13}$

Step 4.1.2

Combine the numerators over the common denominator.

$\frac{66 (3 x^{4} + 13) + 12 x^{3}}{3 x^{4} + 13}$

Step 4.2

Reorder terms.

$\frac{12 x^{3} + 66 (3 x^{4} + 13)}{3 x^{4} + 13}$