Question:
Let \( h(x) = f(\sqrt{g(x)}) \). If \( f'(3) = 6 \), \( g'(3) = 3 \) and \( g(3) = 9 \), then the value of \( h'(3) \) is equal to
Let \( h(x) = f(\sqrt{g(x)}) \). If \( f'(3) = 6 \), \( g'(3) = 3 \) and \( g(3) = 9 \), then the value of \( h'(3) \) is equal to
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Break chain rule step-by-step for nested functions.
Updated On: Apr 21, 2026
- \(1 \)
- \(3 \)
- \(6 \)
- \(9 \)
- \(18 \)
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The Correct Option is B
Solution and Explanation
Concept:
Chain rule:
\[
h'(x) = f'(\sqrt{g(x)}) \cdot \frac{1}{2\sqrt{g(x)}} \cdot g'(x)
\]
Step 1: Substitute values.
\[ \sqrt{g(3)} = \sqrt{9} = 3 \] \[ h'(3) = f'(3) \cdot \frac{1}{2\cdot 3} \cdot g'(3) \] \[ = 6 \cdot \frac{1}{6} \cdot 3 = 3 \]
Step 1: Substitute values.
\[ \sqrt{g(3)} = \sqrt{9} = 3 \] \[ h'(3) = f'(3) \cdot \frac{1}{2\cdot 3} \cdot g'(3) \] \[ = 6 \cdot \frac{1}{6} \cdot 3 = 3 \]
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