Step 1: Understanding the Question:
We are given a composite function \( h(x) \) defined in terms of two other functions, \( f(x) \) and \( g(x) \). We are also given the values of these functions and their derivatives at \( x=1 \). The goal is to find the value of the derivative of \( h(x) \) at \( x=1 \), i.e., \( h'(1) \).
Step 2: Key Formula or Approach:
To find the derivative of \( h(x) \), we need to use the Chain Rule. The function can be written as \( h(x) = [u(x)]^{1/2} \), where the inner function is \( u(x) = 4f(x) + 3g(x) \).
The Chain Rule states that \( h'(x) = \frac{1}{2}[u(x)]^{-1/2} \cdot u'(x) \), which is:
\[ h'(x) = \frac{u'(x)}{2\sqrt{u(x)}} \]
Here, \( u'(x) = \frac{d}{dx}(4f(x) + 3g(x)) = 4f'(x) + 3g'(x) \).
Step 3: Detailed Explanation:
1. Find the general derivative h'(x):
Substitute \( u(x) \) and \( u'(x) \) into the chain rule formula:
\[ h'(x) = \frac{4f'(x) + 3g'(x)}{2\sqrt{4f(x) + 3g(x)}} \]
2. Evaluate h'(1) by substituting x=1:
\[ h'(1) = \frac{4f'(1) + 3g'(1)}{2\sqrt{4f(1) + 3g(1)}} \]
3. Plug in the given values:
Given: \( f(1)=4, g(1)=3, f'(1)=3, g'(1)=4 \).
First, calculate the numerator:
\( 4f'(1) + 3g'(1) = 4(3) + 3(4) = 12 + 12 = 24 \).
Next, calculate the denominator:
\( 2\sqrt{4f(1) + 3g(1)} = 2\sqrt{4(4) + 3(3)} = 2\sqrt{16 + 9} = 2\sqrt{25} = 2(5) = 10 \).
4. Calculate the final value:
\[ h'(1) = \frac{24}{10} = \frac{12}{5} \]
Step 4: Final Answer:
The value of \( h'(1) \) is \( \frac{12}{5} \), which corresponds to option (D).
(Note: The provided answer key in the original file might be inconsistent, but the calculation correctly leads to option D).