Question

Differentiate the following function with respect $x$:
$\cos h^{-1}(2x+1)$

Answer 1

Let $\cos h^{-1}x=y$

$\Rightarrow x=\cos hy$

Differentiate on both sides with respect to x

$\Rightarrow 1=\sin hy\times \frac{dy}{dx}$

$\Rightarrow \frac{dy}{dx}=\frac{1}{\sin hy}$

As $\cos hy=x\Rightarrow \sin hy=\sqrt{x^2-1}$

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

$\therefore \frac{d}{dx}\left(\cos h^{-1}x\right)=\frac{1}{\sqrt{x^2-1}}$

$\Rightarrow \frac{d}{dx}\left(\cos h^{-1}\right(2x+1\left)\right)=\frac{1}{\sqrt{(2x+1{)}^2-1}}\times 2$

$\therefore \frac{d}{dx}\left(\cos h^{-1}\right(2x+1\left)\right)=\frac{1}{\sqrt{x^2+x}}$