Question

Find the second order derivative of the given functions.

$ta{n}^{-1}x$

Answer 1

Let y = $ta{n}^{-1}x$

Differentiating twicely w.r.t. x, we get

$\frac{dy}{dx}=\frac{d}{dx}\left(ta{n}^{-1}x\right)=\frac{1}{1+x^2}and\frac{d^{2}y}{d{x}^2}=\frac{d}{dx}\left(\frac{1}{1+x^2}\right)=\frac{d}{dx}(1+x^2{)}^{-1}=-1(1+x^2{)}^{-2}\frac{d}{dx}(1+x^2)=-1(1+x^2{)}^{-2}(0+2x)=\frac{-2x}{(1+x^2{)}^2}$