Differentiate the following functions with respect to x: a + x a … - Vidyadip

Question

Differentiate the following functions with respect to x:$\frac{\sqrt{\text{a}}+\sqrt{\text{x}}}{\sqrt{\text{a}}-\sqrt{\text{x}}}$

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Answer

We have,$\frac{\text{d}}{\text{dx}}\Big(\frac{\sqrt{\text{a}}+\sqrt{\text{x}}}{\sqrt{\text{a}}-\sqrt{\text{x}}}\Big)$
Using quotient rule, we get
$\frac{(\sqrt{\text{a}}-\sqrt{\text{x}})\frac{\text{d}}{\text{dx}}(\sqrt{\text{a}}+\sqrt{\text{x}})-(\sqrt{\text{a}}+\sqrt{\text{x}})\frac{\text{d}}{\text{dx}}(\sqrt{\text{a}}-\sqrt{\text{x}})}{(\sqrt{\text{a}}-\sqrt{\text{x}})^2}$
$=\frac{(\sqrt{\text{a}}-\sqrt{\text{x}})\times\frac{1}{2\sqrt{\text{x}}}-(\sqrt{\text{a}}+\sqrt{\text{x}})\times\frac{-\text{1}}{\text{2}\sqrt{\text{x}}}}{(\sqrt{\text{a}}-\sqrt{\text{x}})^2}$
$=\frac{\sqrt{\text{a}}-\sqrt{\text{x}}+\sqrt{\text{a}}+\sqrt{\text{x}}}{2\sqrt{\text{x}}(\sqrt{\text{a}}-\sqrt{\text{x}})^2}$
$=\frac{\sqrt{\text{a}}}{\sqrt{\text{x}}(\sqrt{\text{a}}-\sqrt{\text{x}})^2}$

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