Compare three different methods to find a derivative of $f(x) = 2x^3 − x$.

1. Use the definition of a derivative.

   Step 1(a):

   $f'(x)=\lim\limits_{h\rightarrow 0}\frac{(2(x+h)^{3}-(x+h))-(2x^{3}-x)}{h}$

   Step 2(a):

   Expand the numerator.

   Step 3(a):

   Combine like terms to simplify the numerator.

   Step 4(a):

   Factor out an $h$.

   Step 5:

   Simplify the fraction by 'cancelling out' the $h$.

   Step 6:

   Evaluate the limit as $h \rightarrow 0$.

2. Use the Power Rule. Does your answer agree with your answer from part (a)?

   Hint (b):

   If your answers do not agree, you probably made an Algebraic error in part (a).

3. Use your graphing calculator to graph $f^\prime (x)= \frac{f(x+h)-f(x)}{h}$ for $h = 0.01$. Does the graph match that of your answer from part (a)?

   Steps (c):

   Sketch $y=\frac{(2(x+0.01)^{3}-(x+0.01))-(2x^{3}-x)}{0.01}$

   also sketch the $f^\prime (x)$ you found in part (a) and part (b).

   Answer (c):

   The two graphs should be close, but not identical since part (c) uses a finite value for $h$.