Differentiation Question 4
Question 4 - 2024 (29 Jan Shift 1)
Suppose
$f(x)=\frac{\left(2^{x}+2^{-x}\right) \tan x \sqrt{\tan ^{-1}\left(x^{2}-x+1\right)}}{\left(7 x^{2}+3 x+1\right)^{3}}$
Then the value of $f^{\prime}(0)$ is equal to
(1) $\pi$
(2) 0
(3) $\sqrt{\pi}$
(4) $\frac{\pi}{2}$
Show Answer
Answer (3)
Solution
$$ \begin{aligned} & f^{\prime}(0)=\lim _{h \rightarrow 0} \frac{f(h)-f(0)}{h} \\ & =\lim _{h \rightarrow 0} \frac{\left(2^{h}+2^{-h}\right) \tan h \sqrt{\tan ^{-1}\left(h^{2}-h+1\right)}-0}{\left(7 h^{2}+3 h+1\right)^{3} h} \\ & =\sqrt{\pi} \end{aligned} $$
