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

Question

Differentiate the following functions with respect to x:
$\sqrt{\tan^{-1}\big(\frac{\text{x}}{2}\big)}$

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Answer

Let, $\text{y}=\sqrt{\tan^{-1}\big(\frac{\text{x}}{2}\big)}$
$\Rightarrow\ \text{y}=\Big(\tan^{-1}\big(\frac{\text{x}}{2}\big)\Big)^\frac{1}{2}$
Differentiate it with respect to x,
$\frac{\text{dy}}{\text{dx}}=\frac{\text{d}}{\text{dx}}\Big(\tan^{-1}\big(\frac{\text{x}}{2}\big)\Big)^\frac{1}{2}$
$=\frac{1}{2}\Big(\tan^{-1}\frac{\text{x}}{2}\Big)^{\frac{1}{2}-1}\frac{\text{d}}{\text{dx}}\Big(\tan^{-1}\frac{\text{x}}{2}\Big)$
$=\frac{1}{2}\Big(\tan^{-1}\frac{\text{x}}{2}\Big)^{\frac{-1}{2}}\times\frac{1}{1+\big(\frac{\text{x}}{2}\big)^2}\times\frac{\text{d}}{\text{dx}}\big(\frac{\text{x}}{2}\big)$
$=\frac{4}{4\sqrt{\tan^{-1}\big(\frac{\text{x}}{2}\big)}\big(4+\text{x}^2\big)}$
$=\frac{1}{\big(4+\text{x}^2\big)\sqrt{\tan^{-1}\big(\frac{\text{x}}{2}\big)}}$
So,
$\frac{\text{d}}{\text{dx}}\bigg(\sqrt{\tan^{-1}\big(\frac{\text{x}}{2}\big)}\bigg)=\frac{1}{\big(4+\text{x}^2\big)\sqrt{\tan^{-1}\big(\frac{\text{x}}{2}\big)}}$

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