Question

Differentiate

$f\left(x\right)={\log }_e\sqrt{4-x^2}$

Answer 1

$f\left(x\right)=lo{g}_e^{\sqrt{4-x^2}}$

differentiating on both sides

$\therefore f^{\prime }\left(x\right)=\frac{d\left(lo{g}_e^{\sqrt{4-x^2}}\right)}{dx}$

$=\frac{1}{\sqrt{4-x^2}}\frac{d\left(\sqrt{4-x^2}\right)}{dx}$

$(\because \frac{d\left(\ln y\right)}{dx}=\frac{1}{y}\frac{dy}{dx})$

$=\frac{1}{\sqrt{4-x^2}}\left(\frac{-2x}{2\sqrt{4-x^2}}\right)$

$=\frac{-x}{4-x^2}$

$\therefore f^{\prime }\left(x\right)=\frac{-x}{4-x^2}$