Question

If $f(x)=x \tan ^{-1} x$, then $f'(1)$ is equal to :

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

Answer 1

Answer: (B)

$f(x)=x \tan ^{-1} x$
On differentiating both sides w.r.t. $x$, we get
$f'(x) =x \frac{1}{1+x^{2}}+\tan ^{-1} x$
$f'(x)_{x=1} =\frac{1}{1+1^{2}}+\tan ^{-1} 1$
$=\frac{1}{2}+\frac{\pi}{4}$