Question

Find an expression for the derivative of $co\sec h^{-1}x$ in terms of $x$

Answer 1

Let $cosec{h}^{-1}x=y$

$\Rightarrow x\sinh y=1$

Differentiate on both sides

$\Rightarrow x\cosh y\frac{dy}{dx}+\sinh y=0$

$\Rightarrow \frac{dy}{dx}=\frac{-\sinh y}{x\cosh y}$

$\sinh y=\frac{1}{x}\Rightarrow \cosh y=\sqrt{1+{\left(\frac{1}{x}\right)}^2}$

$\Rightarrow \frac{dy}{dx}=\frac{-1}{x}\left(\frac{\left(\frac{1}{x}\right)}{\sqrt{1+{\left(\frac{1}{x}\right)}^2}}\right)$

$\Rightarrow \frac{dy}{dx}=\frac{-1}{x\sqrt{1+x^2}}$