Question

if y=Cos-1[3x + (4root(1-x2))/5] find dy/dx

Answer 1

$\frac{d(Cos^{-1}z)}{dz}=\frac{-1}{\sqrt{1-z^2}},\ \ \ Let\ z=3x+\frac{4}{5}\sqrt{1-x^2}\\\\\frac{dz}{dx}=\frac{d}{dx}[ 3x+\frac{4}{5}\sqrt{1-x^2} ]=(3+\frac{4}{5}*\frac{1}{2*\sqrt{1-x^2}}*(-2x))\\\\.\ \ \ =(\frac{15\sqrt{1-x^2}-4x}{5\sqrt{1-x^2}})\\\\\frac{dy}{dx}=\frac{dy}{dz}*\frac{dz}{dx}=\frac{4x-15\sqrt{1-x^2}}{\sqrt{9-218x^2+209x^4+120x(1-x^2)^{\frac{3}{2}}}}$

Substitute the expressions as per shown formulas and simplify to get the answer as given.