Final answer:
To find the derivatives of the function f(x) = 3x² - x³, we can use the power rule. The first derivative, f'(x), is 6x - 3x². The second derivative, f''(x), is 6 - 6x. The third derivative, f'''(x), is -6. The fourth derivative, f^(4)(x), is 0.
Step-by-step explanation:
To find the derivatives of the function f(x) = 3x² - x³, we can use the power rule. The first derivative, f'(x), can be found by multiplying the coefficient of each term by the exponent of x and then subtracting 1 from the exponent. So, f'(x) = 2(3x) - 3(x²) = 6x - 3x².
The second derivative, f''(x), can be found by applying the power rule again to f'(x). So, f''(x) = 6 - 6x.
The third derivative, f'''(x), can be found by applying the power rule once again to f''(x). So, f'''(x) = -6.
To find f^(4)(x), we continue to differentiate f'''(x) and obtain f^(4)(x) = 0.