Final answer:
The student asked for the fourth derivative of the function f(x) = (2x + 1)^4 at x = 0. After applying the power rule multiple times, the fourth derivative is found to be a constant, which is 192. This value is independent of x, including at x = 0.
Step-by-step explanation:
The student has provided the function f(x) = (2x + 1)^4 and is asking for the fourth derivative of f(x) evaluated at x = 0. To find the fourth derivative, we must first determine the general form of f(x)'s derivatives. Let's start with the first derivative using the power rule, f'(x) = 4(2x + 1)^3 × 2.
Continuing this process, the second derivative is f''(x) = 4 × 3 × 2(2x + 1)^2 × 2, the third derivative is f'''(x) = 4 × 3 × 2 × 2(2x + 1) × 2, and the fourth derivative is f''''(x) = 4 × 3 × 2 × 2 × 2. Since the fourth derivative does not depend on x, its value at x = 0 or any other x-value is simply the constant 192.