Question

Let $f$ and $g$ be differentiable funcitons on $R$, such that $fog$ is the identity funciton. If for some $a,b\in R,g^{\prime }\left(a\right)=5$ and $g\left(a\right)=b$, then $f^{\prime }\left(b\right)$ is equal to :

• A. $\frac{2}{5}$
• B. $5$
• C. $1$
• D. $\frac{1}{5}$

Answer 1

The correct option is D $\frac{1}{5}$

$f\left(g\right(x\left)\right)=x$
$f^{\prime }\left(g\right(x\left)\right)g^{\prime }\left(x\right)=1$
Put$x=a$
$f^{\prime }\left(g\right(a\left)\right)g^{\prime }\left(a\right)=1\Rightarrow f^{\prime }\left(b\right)\times 5=1\Rightarrow f^{\prime }\left(b\right)=\frac{1}{5}$