Final answer:
The derivative f'(a) of the function f(x) = 4x² - 4x³ is found using the power rule of differentiation, resulting in f'(a) = 8a - 12a².
Step-by-step explanation:
To find f'(a) for the function f(x) = 4x² - 4x³, we need to calculate the derivative of the function and then evaluate it at x = a. To derive the function, we use the power rule which states that the derivative of x^n is nx^(n-1). Applying this rule, we differentiate the terms separately:
The derivative of 4x² is 2 * 4x which simplifies to 8x.
The derivative of -4x³ is 3 * -4x² which simplifies to -12x².
Therefore, the derivative f'(x) of the original function is 8x - 12x². To find f'(a), we substitute x with a to get f'(a) = 8a - 12a².