If \(f(u) = u^{3} - 2u\) and \(u = g(x) = \sqrt{x}\), then the value of \((f \circ g)'(x)\) at \(x = 9\) is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\displaystyle\frac{25}{6}\)
By the chain rule, \((f \circ g)'(x) = f'(g(x)) \cdot g'(x)\).
\( \displaystyle f'(u) = 3u^{2} - 2, \quad g'(x) = \frac{1}{2\sqrt{x}} \)
At x = 9, \(u = g(9) = 3\):
\(f'(3) = 27 - 2 = 25\)
\( \displaystyle g'(9) = \frac{1}{2 \times 3} = \frac{1}{6} \)
\( \displaystyle (f \circ g)'(9) = 25 \times \frac{1}{6} = \frac{25}{6} \)