Question

Differentiate the following functions with respect to x :

$\frac{\sqrt{a}+\sqrt{x}}{\sqrt{a}-\sqrt{x}}$

Answer 1

We have,

$\frac{d}{dx}\left(\frac{\sqrt{a}+\sqrt{x}}{\sqrt{a}-\sqrt{x}}\right)$

Using quotient rule,

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

$=\frac{(\sqrt{a}-\sqrt{x})\times \frac{1}{2\sqrt{x}}-(\sqrt{a}+\sqrt{x})\times \frac{-1}{2\sqrt{x}}}{(\sqrt{a}-\sqrt{x}{)}^2}=\frac{\sqrt{a}-\sqrt{x}+\sqrt{a}+\sqrt{x}}{2\sqrt{x}(\sqrt{a}-\sqrt{x}{)}^2}=\frac{\sqrt{a}}{\sqrt{x}(\sqrt{a}-\sqrt{x}{)}^2}$