Calculus Examples

$y = arccoth (\cosh (x))$

Step 1

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

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Step 1.1

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

$\frac{d}{d u} [arccoth (u)] \frac{d}{d x} [\cosh (x)]$

Step 1.2

The derivative of $arccoth (u)$ with respect to $u$ is $\frac{1}{1 - u^{2}}$ .

$\frac{1}{1 - u^{2}} \frac{d}{d x} [\cosh (x)]$

Step 1.3

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

$\frac{1}{1 - \cosh^{2} (x)} \frac{d}{d x} [\cosh (x)]$

Step 2

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

$\frac{1}{1 - \cosh^{2} (x)} \sinh (x)$

Step 3

Combine fractions.

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Step 3.1

Combine $\frac{1}{1 - \cosh^{2} (x)}$ and $\sinh (x)$ .

$\frac{\sinh (x)}{1 - \cosh^{2} (x)}$

Step 3.2

Reorder terms.

$\frac{\sinh (x)}{- \cosh^{2} (x) + 1}$