Differentiate the following functions with respect to x: ^ -1 ( x… - Vidyadip

Question

Differentiate the following functions with respect to x:
$\tan^{-1}\Big(\frac{\sqrt{\text{x}}+\sqrt{\text{a}}}{1-\sqrt{\text{xa}}}\Big)$

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Answer

Let $\text{y}=\tan^{-1}\Big(\frac{\sqrt{\text{x}}+\sqrt{\text{a}}}{1-\sqrt{\text{xa}}}\Big)$$\text{y}=\tan^{-1}\sqrt{\text{x}}+\tan^{-1}\sqrt{\text{a}}$
$\Big[\text{Since},\tan^{-1}\text{x}+\tan^{-1}\text{y}=\tan^{-1}\frac{\text{x}+\text{y}}{1-\text{xy}}\Big]$
Differentiating it with respect to x using chain rule,
$\frac{\text{dy}}{\text{dx}}=\frac{\text{d}}{\text{dx}}(\tan^{-1}\sqrt{\text{x}})+\frac{\text{d}}{\text{dx}}(\tan^{-1}\sqrt{\text{a}})$
$=\frac{1}{1+\big(\sqrt{\text{x}}\big)^2}\times\frac{\text{d}}{\text{dx}}\big(\sqrt{\text{x}}\big)+0$
$=\Big(\frac{1}{1+\text{x}}\Big)\Big(\frac{1}{2\sqrt{\text{x}}}\Big)$
$\frac{\text{dy}}{\text{dx}}=\frac{1}{2\sqrt{\text{x}}(1+\text{x})}$

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