Question:
Let $f$ and $g$ be differentiable functions on $\mathbf{R}$ such that fog is the identity function. If for some $a, b \in \mathbf{R}, g^{\prime}(a)=5$ and $g$ $(a)=b$, then $f^{\prime}(b)$ is equal to:
Correct Option: 1,
Solution:
It is given that functions $f$ and $g$ are differentiable and fog is identity function.
$\therefore \quad(f o g)(x)=x \Rightarrow f(g(x))=x$
Differentiating both sides, we get
$f^{\prime}(g(x)) \cdot g^{\prime}(x)=1$
Now, put $x=a$, then
$f^{\prime}(g(a)) \cdot g^{\prime}(a)=1$
$f^{\prime}(b) \cdot 5=1$
$f^{\prime}(b)=\frac{1}{5}$
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