Question

Let $f$ and $g$ be differentiable functions satistying $g^{\prime }\left(a\right)=2,g\left(a\right)=b$ and $fog=I$ ( identity function) Then $f^{\prime }\left(b\right)$ is equal to-

• A. $2$
• B. $\frac{2}{3}$
• C. $\frac{1}{2}$
• D. None

Answer 1

The correct option is C $\frac{1}{2}$

$fog=x=f\left(g\left(x\right)\right)$

differentiate with respect to x

$f^{\prime }\left(g\left(x\right)\right)\times g^{\prime }\left(x\right)=1$

$f^{\prime }\left(g\left(x\right)\right)=\frac{1}{g^{\prime }\left(x\right)}$

Let $g\left(x\right)=b\Rightarrow x=a$ (from question)

$\Rightarrow f^{\prime }\left(b\right)=\frac{1}{g^{\prime }\left(a\right)}=\frac{1}{2}$