Question

Let $f$ and $g$ be differentiable functions on $R$, such that $fog$ is the identity function. If for some $a,b\in R$, $g\text{'}\left(a\right)=5$ and $g\left(a\right)=b$, then $f\text{'}\left(b\right)$ is equal to

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

Answer 1

The correct option is D

$\frac{1}{5}$

Explanation for correct option

Determining value of $f\text{'}\left(b\right)$

Given data,

$a,b\in R$

$g\text{'}\left(a\right)=5$

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

Given, $fog$ is the identity function. This means,

$$\begin{gathered} \mathrm{fog}=\mathrm{I}\left[\text{I → identity function}\right] \\ \mathrm{fog}=\mathrm{x} \end{gathered}$$

We know that,

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

$$\begin{gathered} \mathrm{f}\text{'}\left(\mathrm{g}\left(\mathrm{x}\right)\right)\times \mathrm{g}\text{'}\left(\mathrm{x}\right)=1\mathbf{}\left[\mathrm{Given},\mathrm{fog}=1\right] \\ \mathrm{f}\text{'}\left(\mathrm{g}\left(\mathrm{x}\right)\right)=\frac{1}{\mathrm{g}\text{'}\left(\mathrm{x}\right)}......\left(1\right) \end{gathered}$$

From the given data,

$$\begin{gathered} g\text{'}\left(a\right)=5 \\ g\text{'}\left(x\right)=5\left(\text{where},a=x\right) \\ g\left(a\right)=b \\ g\left(x\right)=b\left(\text{where},a=x\right) \end{gathered}$$

Substitute the values $g\text{'}\left(x\right)=5$ and $g\left(x\right)=b$ in the Equation $\left(1\right)$:

$$\begin{gathered} f\text{'}\left(g\left(x\right)\right)=\frac{1}{g\text{'}\left(x\right)} \\ f\text{'}\left(b\right)=\frac{1}{5} \end{gathered}$$

Hence, the correct answer is option D.