Final answer:
To find the successive derivatives f'(x), f"(x), f'''(x), and f⁴(x) of the function f(x) = 2x² - x³, we apply the power rule of differentiation. The results are f'(x) = 4x - 3x², f"(x) = 4 - 6x, f'''(x) = -6, and f⁴(x) = 0.
Step-by-step explanation:
Given the function f(x) = 2x² - x³, we can find the successive derivatives as follows:
Finding f'(x):
The first derivative of the function with respect to x, denoted as f'(x), represents the rate of change or the slope of f(x). To find this, we use the power rule of differentiation.
f'(x) = d/dx(2x²) - d/dx(x³) = 4x - 3x²
Finding f"(x):
The second derivative of the function, f"(x), represents the rate of change of the slope, which is the concavity of the graph of f(x). Again, applying the power rule, we get:
f"(x) = d/dx(f'(x)) = d/dx(4x - 3x²) = 4 - 6x
Finding f'''(x):
The third derivative, f'''(x), provides information about the rate of change of the concavity of the graph. Differentially the second derivative gives us:
f'''(x) = d/dx(f"(x)) = d/dx(4 - 6x) = -6
Finding f⁴(x):
The fourth derivative, f⁴(x), would give us the rate of change of the rate of change of the concavity (which in most practical cases is not often used).
f⁴(x) = d/dx(f'''(x)) = d/dx(-6) = 0.
As we continue taking derivatives, we can clearly see that any derivative of order five and above would be zero, since we've reached a constant with the fourth derivative.