Question:
Let f and g be two differentiable functions satisfying $g'(5) = \frac{3}{4}$, $g(5) = 6$ and $g = f^{-1}$. Then $f'(6) =$
Let f and g be two differentiable functions satisfying $g'(5) = \frac{3}{4}$, $g(5) = 6$ and $g = f^{-1}$. Then $f'(6) =$
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Just remember: The slopes of inverse functions at corresponding points are reciprocals of each other.
Updated On: May 12, 2026
- $\frac{1}{2}$
- $\frac{1}{6}$
- $\frac{2}{3}$
- $\frac{4}{3}$
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The Correct Option is D
Solution and Explanation
Step 1: Concept
The derivative of an inverse function is given by the formula: $f'(x) = \frac{1}{g'(f(x))}$ or $g'(x) = \frac{1}{f'(g(x))}$.
Step 2: Meaning
We are given $g = f^{-1}$, which means $f$ and $g$ are inverses. If $g(5) = 6$, then by the property of inverses, $f(6) = 5$.
Step 3: Analysis
Using the derivative of the inverse function formula at $x = 5$ for the function $g$: $g'(5) = \frac{1}{f'(g(5))}$. Substitute the known values: $\frac{3}{4} = \frac{1}{f'(6)}$. Rearranging to find $f'(6)$: $f'(6) = \frac{1}{3/4} = \frac{4}{3}$.
Step 4: Conclusion
The value of $f'(6)$ is $\frac{4}{3}$. Final Answer: (D)
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