Let f and g be differentiable functions on
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:

1. (1) $\frac{1}{5}$

2. (2) 1

3. (3) 5

4. (4) $\frac{2}{5}$

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

$f^{\prime}(g(a)) \cdot g^{\prime}(a)=1$

$f^{\prime}(b) \cdot 5=1$

$f^{\prime}(b)=\frac{1}{5}$