Calculus Examples

$e^{\cosh (3 x)}$

Step 1

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

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as $\cosh (3 x)$ .

$\frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d x} [\cosh (3 x)]$

Step 1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $e$ .

$e^{u_{1}} \frac{d}{d x} [\cosh (3 x)]$

Step 1.3

Replace all occurrences of $u_{1}$ with $\cosh (3 x)$ .

$e^{\cosh (3 x)} \frac{d}{d x} [\cosh (3 x)]$

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 $g (x) = 3 x$ .

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Step 2.1

To apply the Chain Rule, set $u_{2}$ as $3 x$ .

$e^{\cosh (3 x)} (\frac{d}{d u_{2}} [\cosh (u_{2})] \frac{d}{d x} [3 x])$

Step 2.2

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

$e^{\cosh (3 x)} (\sinh (u_{2}) \frac{d}{d x} [3 x])$

Step 2.3

Replace all occurrences of $u_{2}$ with $3 x$ .

$e^{\cosh (3 x)} (\sinh (3 x) \frac{d}{d x} [3 x])$

Step 3

Differentiate.

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Step 3.1

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

$e^{\cosh (3 x)} (\sinh (3 x) (3 \frac{d}{d x} [x]))$

Step 3.2

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

$e^{\cosh (3 x)} (\sinh (3 x) (3 \cdot 1))$

Step 3.3

Simplify the expression.

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Step 3.3.1

Multiply $3$ by $1$ .

$e^{\cosh (3 x)} (\sinh (3 x) \cdot 3)$

Step 3.3.2

Move $3$ to the left of $\sinh (3 x)$ .

$e^{\cosh (3 x)} (3 \cdot \sinh (3 x))$

Step 3.3.3

Reorder the factors of $e^{\cosh (3 x)} (3 \sinh (3 x))$ .

$3 e^{\cosh (3 x)} \sinh (3 x)$