Question

Differentiate the following
(i) ${\tan }^{-1}\left(\frac{2x}{1-x^2}\right)$

Answer 1

Let $y={\tan }^{-1}\left(\frac{2x}{1-x^2}\right)$

Assume , $x=tan\left(\theta \right)$ and ${\tan }^{-1}x=\theta$

So, $y={\tan }^{-1}\left(\frac{2tan\left(\theta \right)}{1-ta{n}^2\left(\theta \right)}\right)$

We know that , $\frac{2\tan \theta }{1-{\tan }^2\theta }=tan2\theta$

Now , $y={\tan }^{-1}tan\left(\theta \right)$

$y=\theta$

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

Differentiate $w.r.t$ $x$

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