Step 1: Understanding the Concept:
This question tests the ability to evaluate derivatives of multiple nested functions.
The notation \( f'(h'(g'(x))) \) means we must find the derivative of \( f \), and evaluate it at a value given by the derivative of \( h \).
The argument of \( h' \) is itself the derivative of \( g(x) \).
In such nested expressions, it is critical to observe whether any of the intermediate derivatives are constant functions.
If a derivative \( \phi'(x) \) is a constant, its value does not change regardless of what is passed into it. This often significantly simplifies "stacked" functional problems.
Step 2: Key Formula or Approach:
We evaluate from the inside out:
1. Find \( g'(x) \).
2. Find \( h'(g'(x)) \).
3. Find \( f'(h'(g'(x))) \).
Derivative of \( \sqrt{u} \) is \( \frac{1}{2\sqrt{u}} \cdot \frac{du}{dx} \).
Derivative of \( ax + b \) is \( a \).
Step 3: Detailed Explanation:
Let's start by calculating \( h'(x) \) because \( h(x) \) is a simple linear function.
Given: \( h(x) = 2x - 3 \).
Differentiating with respect to \( x \):
\[ h'(x) = 2 \]
Note that \( h'(x) \) is a constant. Its value is always 2, regardless of the input.
Therefore, \( h'(g'(x)) = 2 \), no matter how complex the expression for \( g'(x) \) might be.
We effectively don't even need to calculate \( g'(x) \) to solve this problem.
The expression we need to evaluate now is:
\[ f'(h'(g'(x))) = f'(2) \]
Now, let's find the derivative of \( f(x) \).
Given: \( f(x) = \sqrt{x^2 + 1} \).
Rewrite as: \( f(x) = (x^2 + 1)^{1/2} \).
Using the chain rule:
\[ f'(x) = \frac{1}{2}(x^2 + 1)^{-1/2} \cdot \frac{d}{dx}(x^2 + 1) \]
\[ f'(x) = \frac{1}{2\sqrt{x^2 + 1}} \cdot (2x) \]
\[ f'(x) = \frac{x}{\sqrt{x^2 + 1}} \]
Finally, evaluate \( f'(x) \) at the input we found earlier, which is 2:
\[ f'(2) = \frac{2}{\sqrt{2^2 + 1}} \]
\[ f'(2) = \frac{2}{\sqrt{4 + 1}} \]
\[ f'(2) = \frac{2}{\sqrt{5}} \]
This matches Option (A).
Step 4: Final Answer:
Because the derivative of \( h(x) \) was a constant 2, the complex inner function \( g(x) \) became irrelevant. Evaluating the derivative of \( f(x) \) at 2 yielded \( 2/\sqrt{5} \). This corresponds to Option (A).