Let $ f(x) = x^3 $.

(a) Estimate the values of $ f'(0) $, $ f'(\frac{1}{2}) $, $ f'(1) $, $ f'(2) $, and $ f'(3) $ by using a graphing device to zoom in on the graph of $ f $.

(b) Use symmetry to deduce the values of $ f'(-\frac{1}{2}) $, $ f'(-1) $, $ f'(-2) $, and $ f'(-3) $.

(c) Use the values from parts (a) and (b) to graph $ f' $.

(d) Guess a formula for $ f'(x) $.

(e) Use the definition of derivative to prove that your guess in part (d) is correct.