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First Principle of Differentiation

Prove that d...

Question

Prove that
$\frac{d}{d x} (cot h^{- 1} x) = \frac{- 1}{(x^{2} - 1)}$

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Solution

Let $c o t h^{- 1} x = y$

$\Rightarrow x s i n h y = c o s h y$

Differentiate on both sides

$\Rightarrow x c o s h y \frac{d y}{d x} + s i n h y = s i n h y \frac{d y}{d x}$

$\Rightarrow \frac{d y}{d x} = \frac{s i n h y}{s i n h y - x c o s h y}$

$\Rightarrow \frac{d y}{d x} = \frac{1}{1 - x c o t h y}$

$\Rightarrow \frac{d y}{d x} = \frac{- 1}{x^{2} - 1}$

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