If y=1/(1-x^2)^(1/2) then find dy/dx

Question : If y=1/(1-x^2)^(1/2) then find dy/dx

Solution :

Given

\[y=\frac{1}{\sqrt{1-x^2}}\]

is a rational function.

So to find it’s derivative, we shall use formula of derivative for a rational function

\[\frac{d}{dx}(\frac{u}{v})=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}\]

Where

\[u =1 , v =\sqrt{1-x^2}\]

then

\[\frac{dy}{dx}=\frac{d}{dx}(\frac{u}{v})=\frac{\sqrt{1-x^2}\frac{d(1)}{dx}-1\frac{d(\sqrt{1-x^2})}{dx}}{1-x^2}\]

\[\frac{dy}{dx}=\frac{d}{dx}(\frac{u}{v})=\frac{\sqrt{1-x^2}*0-1*\left(-\frac{x}{\sqrt{1-x^2}}\right)}{1-x^2}\]

\[\frac{dy}{dx}=\frac{x}{\sqrt{1-x^2}}\]