differentiate the following function with respect to x

(1) cos^-1(2x (sqrt(1-x^2))),1/root(2) < x <1

y=cos^-12xv(1-x²)

dy/dx=-1/v{1-4x²(1-x²)}.2{x.(-2x)/2v(1-x²)+v(1-x²)}

=-1/v(4x⁴-4x²+1).2[-x²/v(1-x²) +v(1-x²)]

={-2/v(2x²-1)² }.{-x²+1-x²}/v(1-x²)

=-2(-2x²+1)/{(2x²-1).v(1-x²)}

=2(2x²-1)/{(2x²-1).v(1-x²) }

=2/v(1-x²)

Thanks & regards

Rinkoo Gupta

AskIITians Faculty