Question

Find derivative of following function with respect to x : $co{s}^{-1}\left(\frac{2x}{1-x^2}\right),wherexϵ(-1,1)$.

Answer 1

$\frac{d}{dx}co{s}^{-1}\left(\frac{2x}{1-x^2}\right)$

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

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

$=\frac{-1}{\sqrt{1-{\left(\frac{2x}{1-x^2}\right)}^2}}.[-1.2x.(1-x^2{)}^{-2}.(-2x)+2(1-x^2{)}^{-1}]$

$=\frac{-1}{\sqrt{1-{\left(\frac{2x}{1-x^2}\right)}^2}}.\left[4x^2\right(1-x^2{)}^{-2}+2(1-x^2{)}^{-1}]$