Differentiate ^ -1 [ 1+x^ 2 -1 x ] with respect to x. - Vidyadip

Question

Differentiate $\tan^{-1 }\Bigg[\frac{\sqrt{\text{1+x}^{2}-1}}{\text{x}}\Bigg]$ with respect to $x.$

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Answer

Let $x = \tan \theta$
$\therefore$ Given expression becomes $y = \tan^{-1} \Bigg(\frac{\sec\theta-1}{\tan\theta}\Bigg)=\tan^{-1}\Bigg(\frac{1-\cos\theta}{\sin\theta}\Bigg)$
$\therefore\text{y = tan}^{-1}\Bigg(\tan\frac{\theta}{2}\Bigg)=\frac{1}{2}\theta=\frac{1}{2}\tan^{-1}\text{x}$
$\therefore\frac{\text{dy}}{\text{dx}}=\frac{1}{\text{2(1+x}^{2})}$.

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