Question

Let $f:R\to R$ be a one-one onto differentiable function, such that f(2) = 1 and f '(2) = 3. Then the value of ${\left(\frac{d}{\mathrm{dx}}\left(f^{-1}\left(x\right)\right)\right)}_{x=1}$ is

• A. 1
• B. $\frac{1}{2}$
• C. $\frac{1}{3}$
• D. 3

Answer 1

The correct option is C $\frac{1}{3}$

$\mathrm{Let}{f}^{-1}\left(x\right)=g\left(x\right)\therefore f\left(g\right(x\left)\right)=x\mathrm{Differentiating}\mathrm{both}\mathrm{sides}w.r.t.x,\mathrm{we}\mathrm{get}\therefore f\text{'}\left(g\right(x\left)\right)g\text{'}\left(x\right)=1\Rightarrow g\text{'}\left(x\right)=\frac{1}{f\text{'}\left(g\right(x\left)\right)}\therefore g\text{'}\left(1\right)=\frac{1}{f\text{'}\left(g\right(1\left)\right)}=\frac{1}{f\text{'}\left(2\right)}=\frac{1}{3}$