Question

Find the derivative of $y=\frac{x}{\sqrt{a^2-x^2}}$

Answer 1

$y=\frac{x}{\sqrt{a^2-x^2}}$ (By division Rule of differentiation)

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

$=\frac{\sqrt{a^2-x^2}\left(1\right)-\frac{1}{2\sqrt{a^2-x^2}}}{(a^2-x^2)}(0-2x)$

$=\frac{\sqrt{a^2-x^2}+\frac{2x^2}{2\sqrt{a^2-x^2}}}{(a^2-x^2)}$

$=\frac{(a^2-x^2)+x^2}{\sqrt{a^2-x^2}(a^2-x^2)}$

$=\frac{a^2}{(a^2-x^2{)}^{\frac{3}{2}}}$.