Calculus Examples

$y = \ln (\cosh (z))$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d z} (y) = \frac{d}{d z} (\ln (\cosh (z)))$

Step 2

The derivative of $y$ with respect to $z$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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Step 3.1.1

To apply the Chain Rule, set $u$ as $\cosh (z)$ .

$\frac{d}{d u} [\ln (u)] \frac{d}{d z} [\cosh (z)]$

Step 3.1.2

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

$\frac{1}{u} \frac{d}{d z} [\cosh (z)]$

Step 3.1.3

Replace all occurrences of $u$ with $\cosh (z)$ .

$\frac{1}{\cosh (z)} \frac{d}{d z} [\cosh (z)]$

Step 3.2

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

$\frac{1}{\cosh (z)} \sinh (z)$

Step 3.3

Combine $\frac{1}{\cosh (z)}$ and $\sinh (z)$ .

$\frac{\sinh (z)}{\cosh (z)}$

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = \frac{\sinh (z)}{\cosh (z)}$

Step 5

Replace $y'$ with $\frac{d y}{d z}$ .

$\frac{d y}{d z} = \frac{\sinh (z)}{\cosh (z)}$