Question

Differentiate $\log (1+x^2)$ with respect ${\tan }^{-1}x$.

• A. $2x$
• B. $x^2$
• C. $x$
• D. $x^3$

Answer 1

The correct option is A $2x$

Let $y=\log (1+x^2)$

$y={\tan }^{-1}\left(x\right)$

To find $\frac{dy}{dz}$

first we find $\frac{dy}{dx}=\frac{d}{dx}\left(\log \right(1+x^2)=\frac{2x}{1+x^2}$

Next we find

$\frac{dz}{dx}=\frac{d}{dx}\left({\tan }^{-1}\right(x\left)\right)=\frac{1}{1+x^2}$

Now $\frac{dy}{dz}=\frac{\frac{dy}{dx}}{\frac{dz}{dx}}=\frac{\frac{2x}{1+x^2}}{\frac{1}{1+x^2}}=2x$