Question

Differentiate the following functions with respect to $x$:

${\cos }^{-1}\left\{2x\sqrt{1-x^2}\right\},\frac{1}{\sqrt{2}}<x<1$.

Answer 1

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

${\cos }^{-1}(2\cos \theta .\sin \theta )$

$\frac{\pi }{2}-{\sin }^{-1}\left(\sin 2\theta \right)$

$\frac{\pi }{2}-2\theta$

$\frac{\pi }{2}-2{\cos }^{-1}\left(x\right)$

derivate

$-2\left(\frac{-1}{\sqrt{1-x^2}}\right)$

$\frac{2}{\sqrt{1-x^2}}$