Question

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

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

Answer 1

The correct option is A

$\frac{1}{2}$

Explanation for correct option:

Finding the value of $f\text{'}\left(b\right)$ using the given function:

Given data

$g\text{'}\left(a\right)=2$

$g\left(a\right)=b$

$fog=I\left(\text{Identity function}\right).$

Consider the given equation as,

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

Then,

$$\begin{gathered} f\text{'}\left(g\left(x\right)\right)\left(g\text{'}\left(x\right)\right)=1 \\ f\text{'}\left(g\left(x\right)\right)=\frac{1}{g\text{'}\left(x\right)}......\left(1\right) \end{gathered}$$

From the given data

$$\begin{gathered} g\text{'}\left(a\right)=2 \\ g\text{'}\left(x\right)=2\left(\text{where}x=a\right) \\ g\left(a\right)=b \\ g\left(x\right)=b\left(\text{where}x=a\right) \end{gathered}$$

Substitute the above value in the Equation (1)

$$\begin{gathered} f\text{'}\left(g\left(x\right)\right)=\frac{1}{g\text{'}\left(x\right)} \\ f\text{'}\left(b\right)=\frac{1}{2} \end{gathered}$$

Hence, the correct answer is Option (A).