Q109 / 160 · TS EAMCET 2021 (06 Aug Shift 1) · Mathematics
Let $f$ and $g$ be two differentiable functions satisfying $g^{\prime}(5)=\frac{3}{4}, g(5)=6$ and $g=f^{-1}$. Then $f^{\prime}(6)$ is equal to
- A$\frac{1}{2}$
- B$\frac{1}{6}$
- C$\frac{2}{3}$
- Correct answerD$\frac{4}{3}$
Solution
Given, $g(x)=f^{-1}(x)$
$$
\Rightarrow \quad f[g(x)]=x
$$
Differentiating $f^{\prime}[g(x)] \cdot g^{\prime}(x)=1$
$$
f^{\prime}[g(x)]=\frac{1}{g^{\prime}(x)}
$$
Putting $x=5$, we get
$$
f^{\prime}[g(5)]=\frac{1}{g^{\prime}(5)} \Rightarrow f^{\prime}(6)=\frac{1}{\frac{3}{4}}=\frac{4}{3}
$$
Hence, the correct answer is option: 'D'.