Final answer:
The second derivative of the function f(x) = 15x^2 − x^3 represents the rate of change of the first derivative. At x = −1, the second derivative is calculated to be 36. Hence, the rate of change of f '(x) at the point (−1, 16) is 36.
Step-by-step explanation:
To determine the rate of change of f '(x) at the point (−1, 16), we need to find the second derivative, which represents the rate of change of the first derivative. The function given is f(x) = 15x2 − x3. First, we calculate the first derivative, f '(x), which reveals how the function's slope changes:
f '(x) = d/dx (15x2 − x3)
= 30x − 3x2.
Next, we find the second derivative, f ''(x), to see how the first derivative changes:
f ''(x) = d/dx (30x − 3x2)
= 30 − 6x.
The rate of change of f '(x) at x = −1 is f ''(−1) = 30 − 6(−1) = 36. Therefore, the rate of change of f '(x) at the point (−1, 16) is 36.