Question

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

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

Answer 1

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

The given equation is;

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

Differentiating once w.r.t we get,

$\Rightarrow fo{g}^{\prime }=f^{\prime }\left(g\right(x\left)\right).g^{\prime }\left(x\right)$

Alse $fo{g}^{\prime }=1$

$\Rightarrow fo{g}^{\prime }\left(a\right)=f^{\prime }\left(g\right(a\left)\right).g^{\prime }\left(a\right)=1$

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

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