Suppose f(x) = (2^x+2^−x)tanx√tan^−1(x^2−x+1) / √(7x^2+3x+1)^3,

Question

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) \(\frac{\pi}{2}\)

Answer 1

Correct option is (3) \(\sqrt{\pi}\)

\(f^{\prime}(0)=\lim\limits _{h \rightarrow 0} \frac{f(h)-f(0)}{h}\) 

\(=\lim\limits _{h \rightarrow 0} \frac{\left(2^{h}+2^{-h}\right) \tan h \sqrt{\tan ^{-1}\left(h^{2}-h+1\right)}-0}{\left(7 h^{2}+3 h+1\right)^{3} h}\)

\(=\sqrt{\pi}\)