A

To find the derivative of the function \( (Ax + B)^n \), we will use the chain rule from calculus.

Step 1: Identify the outer function and inner function.
Let \( u = Ax + B \). Then the function can be rewritten as \( u^n \).

Step 2: Apply the chain rule.
According to the chain rule, \( \frac{d}{dx}[u^n] = n u^{n-1} \frac{du}{dx} \).

Step 3: Calculate \( \frac{du}{dx} \).
For our inner function, \( u = Ax + B \), the derivative is:
\( \frac{du}{dx} = A \).

Step 4: Combine the results.
Now, substituting back into the chain rule:
\( \frac{d}{dx}[(Ax + B)^n] = n(Ax + B)^{n-1} \cdot A = nA(Ax + B)^{n-1} \).

Thus the final answer is \( nA(Ax + B)^{n-1} \), corresponding to option A.