Calculus Examples

$4 \sinh (4 x) + 6 \cosh (4 x)$

Step 1

By the Sum Rule, the derivative of $4 \sinh (4 x) + 6 \cosh (4 x)$ with respect to $x$ is $\frac{d}{d x} [4 \sinh (4 x)] + \frac{d}{d x} [6 \cosh (4 x)]$ .

$\frac{d}{d x} [4 \sinh (4 x)] + \frac{d}{d x} [6 \cosh (4 x)]$

Step 2

Evaluate $\frac{d}{d x} [4 \sinh (4 x)]$ .

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Step 2.1

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

$4 \frac{d}{d x} [\sinh (4 x)] + \frac{d}{d x} [6 \cosh (4 x)]$

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 $f (x) = \sinh (x)$ and $g (x) = 4 x$ .

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Step 2.2.1

To apply the Chain Rule, set $u_{1}$ as $4 x$ .

$4 (\frac{d}{d u_{1}} [\sinh (u_{1})] \frac{d}{d x} [4 x]) + \frac{d}{d x} [6 \cosh (4 x)]$

Step 2.2.2

The derivative of $\sinh (u_{1})$ with respect to $u_{1}$ is $\cosh (u_{1})$ .

$4 (\cosh (u_{1}) \frac{d}{d x} [4 x]) + \frac{d}{d x} [6 \cosh (4 x)]$

Step 2.2.3

Replace all occurrences of $u_{1}$ with $4 x$ .

$4 (\cosh (4 x) \frac{d}{d x} [4 x]) + \frac{d}{d x} [6 \cosh (4 x)]$

Step 2.3

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

$4 (\cosh (4 x) (4 \frac{d}{d x} [x])) + \frac{d}{d x} [6 \cosh (4 x)]$

Step 2.4

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

$4 (\cosh (4 x) (4 \cdot 1)) + \frac{d}{d x} [6 \cosh (4 x)]$

Step 2.5

Multiply $4$ by $1$ .

$4 (\cosh (4 x) \cdot 4) + \frac{d}{d x} [6 \cosh (4 x)]$

Step 2.6

Move $4$ to the left of $\cosh (4 x)$ .

$4 (4 \cdot \cosh (4 x)) + \frac{d}{d x} [6 \cosh (4 x)]$

Step 2.7

Multiply $4$ by $4$ .

$16 \cosh (4 x) + \frac{d}{d x} [6 \cosh (4 x)]$

Step 3

Evaluate $\frac{d}{d x} [6 \cosh (4 x)]$ .

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Step 3.1

Since $6$ is constant with respect to $x$ , the derivative of $6 \cosh (4 x)$ with respect to $x$ is $6 \frac{d}{d x} [\cosh (4 x)]$ .

$16 \cosh (4 x) + 6 \frac{d}{d x} [\cosh (4 x)]$

Step 3.2

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 $g (x) = 4 x$ .

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Step 3.2.1

To apply the Chain Rule, set $u_{2}$ as $4 x$ .

$16 \cosh (4 x) + 6 (\frac{d}{d u_{2}} [\cosh (u_{2})] \frac{d}{d x} [4 x])$

Step 3.2.2

The derivative of $\cosh (u_{2})$ with respect to $u_{2}$ is $\sinh (u_{2})$ .

$16 \cosh (4 x) + 6 (\sinh (u_{2}) \frac{d}{d x} [4 x])$

Step 3.2.3

Replace all occurrences of $u_{2}$ with $4 x$ .

$16 \cosh (4 x) + 6 (\sinh (4 x) \frac{d}{d x} [4 x])$

Step 3.3

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

$16 \cosh (4 x) + 6 (\sinh (4 x) (4 \frac{d}{d x} [x]))$

Step 3.4

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

$16 \cosh (4 x) + 6 (\sinh (4 x) (4 \cdot 1))$

Step 3.5

Multiply $4$ by $1$ .

$16 \cosh (4 x) + 6 (\sinh (4 x) \cdot 4)$

Step 3.6

Move $4$ to the left of $\sinh (4 x)$ .

$16 \cosh (4 x) + 6 (4 \cdot \sinh (4 x))$

Step 3.7

Multiply $4$ by $6$ .

$16 \cosh (4 x) + 24 \sinh (4 x)$

Step 4

Reorder terms.

$24 \sinh (4 x) + 16 \cosh (4 x)$