Differentiate the following w. r. t. x. ^ -1 (1 x^2 ) - Vidyadip

Question

Differentiate the following w. r. t. x.$\cot ^{-1}\left(\frac{1}{x^2}\right)$

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Answer

Let $y=\cot ^{-1}\left(\frac{1}{x^2}\right)=\tan ^{-1}\left(x^2\right)$
Differentiate $w, r . t . x$.
$
\begin{aligned}
\frac{d y}{d x} & =\frac{d}{d x}\left(\tan ^{-1}\left(x^2\right)\right) \\
& =\frac{1}{1+\left(x^2\right)^2} \cdot \frac{d}{d x}\left(x^2\right) \\
\therefore \quad \frac{d y}{d x} & =\frac{2 x}{1+x^4}
\end{aligned}
$

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