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

• A. \(1 \)
• B. \(3 \)
• C. \(6 \)
• D. \(9 \)
• E. \(18 \)

Hint:

Break chain rule step-by-step for nested functions.

Answer 1

The Correct Option is B

Note: The OCR'd question is missing a square root. The image shows \(h(x) = f(\sqrt{g(x)})\). We will proceed with this corrected version.
Step 1: Understanding the Concept:
This problem requires the application of the chain rule for differentiation to a composite function. The function `h` is a composition of `f`, the square root function, and `g`.
Step 2: Key Formula or Approach:
The chain rule states that if \(h(x) = u(v(w(x)))\), then \(h'(x) = u'(v(w(x))) \cdot v'(w(x)) \cdot w'(x)\). In our case, let \(u(y) = f(y)\) and \(v(z) = \sqrt{z}\) and \(w(x) = g(x)\). So \(h(x) = f(\sqrt{g(x)})\). The derivative is: \[ h'(x) = f'(\sqrt{g(x)}) \cdot \frac{d}{dx}(\sqrt{g(x)}) \] And \(\frac{d}{dx}(\sqrt{g(x)}) = \frac{1}{2\sqrt{g(x)}} \cdot g'(x)\). Combining these, we get: \[ h'(x) = f'(\sqrt{g(x)}) \cdot \frac{g'(x)}{2\sqrt{g(x)}} \] Step 3: Detailed Explanation:
We need to find \(h'(3)\). We use the formula derived above and substitute \(x=3\): \[ h'(3) = f'(\sqrt{g(3)}) \cdot \frac{g'(3)}{2\sqrt{g(3)}} \] We are given the following values:
\(g(3) = 9\)
\(g'(3) = 3\)
Substitute these into the expression: \[ h'(3) = f'(\sqrt{9}) \cdot \frac{3}{2\sqrt{9}} \] \[ h'(3) = f'(3) \cdot \frac{3}{2 \cdot 3} = f'(3) \cdot \frac{1}{2} \] We are also given that \(f'(3) = 6\). Substitute this value: \[ h'(3) = 6 \cdot \frac{1}{2} = 3 \] Step 4: Final Answer:
The value of \(h'(3)\) is 3.