Question

$f$ and $g$ are two differentiable function and $fog=I$ (identity function) and $g\text{'}\left(a\right)=2$, $g\left(a\right)=b$, then $f\text{'}\left(b\right)=$

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

Answer 1

The correct option is B

$\frac{1}{2}$

Step 1. Differentiate $fog=x$ with respect to $x$:

$fog=x$

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

$\Rightarrow$$f\text{'}\left(g\right(x\left)\right)\times g\text{'}\left(x\right)=1$

$\Rightarrow$ $f\text{'}\left(g\right(x\left)\right)=\frac{1}{g\text{'}\left(x\right)}$ $\dots ..\left(1\right)$

Let $g\left(x\right)=b$

$\Rightarrow$ $x=a$ $\left[\because g\left(a\right)=b\right]$

Step 2. Put the value of $g\left(x\right)$ and $x$ in equation(1):

$\Rightarrow$ $f\text{'}\left(b\right)=\frac{1}{g\text{'}\left(a\right)}$

$=\frac{\mathbf{1}}{\mathbf{2}}$

Hence, Option ‘B’ is Correct.