Step 1: Understanding the Concept:
The core of this problem lies in the differentiation of composite functions, often referred to as "function of a function."
In mathematical analysis, when a function depends on another variable that is itself a function of \( x \), we cannot apply standard differentiation rules directly to the innermost variable.
Instead, we must peel back the layers of the function one by one, from the outermost to the innermost.
Here, the function \( g(x) \) is built through multiple stages: an innermost function \( h(x) \), followed by a linear transformation \( 4h(x)+3 \), then the application of function \( f \), and finally a squaring operation.
The Chain Rule provides the necessary framework to handle these nested dependencies by multiplying the derivatives of each successive layer.
Step 2: Key Formula or Approach:
The primary tool used here is the Chain Rule for differentiation:
\[ \frac{d}{dx} [F(G(x))] = F'(G(x)) \cdot G'(x) \]
Additionally, we utilize the power rule for the square:
\[ \frac{d}{dx} [u(x)]^n = n \cdot [u(x)]^{n-1} \cdot u'(x) \]
The problem can be simplified significantly by substituting the expression for \( f(x) \) into \( g(x) \) before differentiating, which reduces the number of "links" in our chain.
Step 3: Detailed Explanation:
Let us first simplify the expression for the function \( g(x) \) to make the differentiation process more manageable.
We are given \( f(x) = x - 1 \). This means whatever argument we provide to \( f \), the output will be that argument minus one.
The argument provided to \( f \) in the definition of \( g(x) \) is \( [4h(x) + 3] \).
Applying the definition of \( f \):
\[ f[4h(x) + 3] = (4h(x) + 3) - 1 \]
\[ f[4h(x) + 3] = 4h(x) + 2 \]
Now, we can rewrite \( g(x) \) using this simplified internal term:
\[ g(x) = (4h(x) + 2)^2 \]
To find \( g'(x) \), we differentiate this expression with respect to \( x \).
According to the power rule and chain rule, if we have \( g(x) = [u(x)]^2 \), then \( g'(x) = 2 \cdot u(x) \cdot u'(x) \).
In our case, let \( u(x) = 4h(x) + 2 \).
The derivative of \( u(x) \) is:
\[ u'(x) = \frac{d}{dx} (4h(x) + 2) = 4h'(x) + 0 = 4h'(x) \]
Substituting these back into the formula for \( g'(x) \):
\[ g'(x) = 2 \cdot (4h(x) + 2) \cdot (4h'(x)) \]
\[ g'(x) = 8(4h(x) + 2)h'(x) \]
Now, we are required to find the value of the derivative at the point \( x = 1 \).
Substituting \( x = 1 \) into the derivative formula:
\[ g'(1) = 8(4h(1) + 2)h'(1) \]
From the problem statement, we have the values: \( h(1) = 4 \) and \( h'(1) = -2 \).
Plugging these values into our expression:
\[ g'(1) = 8(4(4) + 2)(-2) \]
First, calculate the value inside the parentheses:
\[ 4 \times 4 = 16 \]
\[ 16 + 2 = 18 \]
Now, multiply the remaining terms:
\[ g'(1) = 8 \times 18 \times (-2) \]
\[ 8 \times 18 = 144 \]
\[ 144 \times (-2) = -288 \]
The final numerical result is \(-288\).
Step 4: Final Answer:
By evaluating the nested composition of functions and applying the Chain Rule, we determined that \( g'(1) = -288 \). This matches Option (A).