Calculus Examples

$g (x) = {(\ln (x) + 7)}^{5}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{5}$ and $g (x) = \ln (x) + 7$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $\ln (x) + 7$ .

$\frac{d}{d u} [u^{5}] \frac{d}{d x} [\ln (x) + 7]$

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 5$ .

$5 u^{4} \frac{d}{d x} [\ln (x) + 7]$

Step 1.3

Replace all occurrences of $u$ with $\ln (x) + 7$ .

$5 {(\ln (x) + 7)}^{4} \frac{d}{d x} [\ln (x) + 7]$

Step 2

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

$5 {(\ln (x) + 7)}^{4} (\frac{d}{d x} [\ln (x)] + \frac{d}{d x} [7])$

Step 3

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

$5 {(\ln (x) + 7)}^{4} (\frac{1}{x} + \frac{d}{d x} [7])$

Step 4

Differentiate using the Constant Rule.

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Step 4.1

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$5 {(\ln (x) + 7)}^{4} (\frac{1}{x} + 0)$

Step 4.2

Combine fractions.

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Step 4.2.1

Add $\frac{1}{x}$ and $0$ .

$5 {(\ln (x) + 7)}^{4} \frac{1}{x}$

Step 4.2.2

Combine $\frac{1}{x}$ and $5$ .

$\frac{5}{x} {(\ln (x) + 7)}^{4}$

Step 4.2.3

Combine $\frac{5}{x}$ and ${(\ln (x) + 7)}^{4}$ .

$\frac{5 {(\ln (x) + 7)}^{4}}{x}$