Calculus Examples

$4 \cosh (\frac{x}{3} - \ln (3))$

Step 1

Since $4$ is constant with respect to $x$ , the derivative of $4 \cosh (\frac{x}{3} - \ln (3))$ with respect to $x$ is $4 \frac{d}{d x} [\cosh (\frac{x}{3} - \ln (3))]$ .

$4 \frac{d}{d x} [\cosh (\frac{x}{3} - \ln (3))]$

Step 2

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

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Step 2.1

To apply the Chain Rule, set $u$ as $\frac{x}{3} - \ln (3)$ .

$4 (\frac{d}{d u} [\cosh (u)] \frac{d}{d x} [\frac{x}{3} - \ln (3)])$

Step 2.2

The derivative of $\cosh (u)$ with respect to $u$ is $\sinh (u)$ .

$4 (\sinh (u) \frac{d}{d x} [\frac{x}{3} - \ln (3)])$

Step 2.3

Replace all occurrences of $u$ with $\frac{x}{3} - \ln (3)$ .

$4 (\sinh (\frac{x}{3} - \ln (3)) \frac{d}{d x} [\frac{x}{3} - \ln (3)])$

Step 3

Differentiate.

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Step 3.1

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

$4 \sinh (\frac{x}{3} - \ln (3)) (\frac{d}{d x} [\frac{x}{3}] + \frac{d}{d x} [- \ln (3)])$

Step 3.2

Since $\frac{1}{3}$ is constant with respect to $x$ , the derivative of $\frac{x}{3}$ with respect to $x$ is $\frac{1}{3} \frac{d}{d x} [x]$ .

$4 \sinh (\frac{x}{3} - \ln (3)) (\frac{1}{3} \frac{d}{d x} [x] + \frac{d}{d x} [- \ln (3)])$

Step 3.3

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

$4 \sinh (\frac{x}{3} - \ln (3)) (\frac{1}{3} \cdot 1 + \frac{d}{d x} [- \ln (3)])$

Step 3.4

Multiply $\frac{1}{3}$ by $1$ .

$4 \sinh (\frac{x}{3} - \ln (3)) (\frac{1}{3} + \frac{d}{d x} [- \ln (3)])$

Step 3.5

Since $- \ln (3)$ is constant with respect to $x$ , the derivative of $- \ln (3)$ with respect to $x$ is $0$ .

$4 \sinh (\frac{x}{3} - \ln (3)) (\frac{1}{3} + 0)$

Step 3.6

Combine fractions.

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Step 3.6.1

Add $\frac{1}{3}$ and $0$ .

$4 \sinh (\frac{x}{3} - \ln (3)) \frac{1}{3}$

Step 3.6.2

Combine $\frac{1}{3}$ and $4$ .

$\frac{4}{3} \sinh (\frac{x}{3} - \ln (3))$

Step 3.6.3

Combine $\frac{4}{3}$ and $\sinh (\frac{x}{3} - \ln (3))$ .

$\frac{4 \sinh (\frac{x}{3} - \ln (3))}{3}$