Question

If $y =\cot ^{-1}\left( x ^{2}\right)$, then the value of $\frac{ dy }{ dx }$ is equal to:

• A. $\frac{2 x}{1+x^{4}}$
• B. $\frac{2 x}{\sqrt{1+4 x}}$
• C. $\frac{-2 x}{1+x^{4}}$
• D. $\frac{-2 x}{\sqrt{1+x^{2}}}$

Answer 1

Answer: (C)

Let $y=\cot ^{-1}\left(x^{2}\right) $
$\Rightarrow \cot y=x^{2}$
Diff both side, w.r.t. '$x$' ;
$-cosec\,{}^{2} y \cdot \frac{d y}{d x}=2 x$
$\frac{d y}{d x}=\frac{2 x}{-cosec\,{}^{2} y}=\frac{2 x}{-\left(1+\cot ^{2} y\right)}=\frac{-2 x}{\left(x^{4}+1\right)}$