Chapter 2.2, Problem 18E

Let $f\left(x\right)=x^3$.

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

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

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

(d) Guess a formula for $f\prime \left(x\right)$.

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