Discussion
Let \(h(x) = f\left(\sqrt{g(x)}\right)\). If \(f'(3) = 6, g'(3) = 3\) and \(g(3) = 9\), then the value of \(h'(3)\) is equal to
Answer: B
\(h'(3) = \) 3.
By the chain rule, \(h'(x) = f'\left(\sqrt{g(x)}\right)\cdot\frac{g'(x)}{2\sqrt{g(x)}}\). At \(x = 3\), \(\sqrt{g(3)} = 3\):
\[ \begin{aligned} h'(3) &= f'(3)\cdot\frac{g'(3)}{2 \times 3} \\ &= 6 \times \frac{3}{6} = 3 \end{aligned} \]
18 multiplies \(f'(3)\) by \(g'(3)\) and forgets the derivative of the square root; 9 is \(g(3)\); 6 is \(f'(3)\) alone.
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