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

1. Use the definition of a derivative.

   Step 1(a):

   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 $h→0$.

2. Use the Power Rule. Do your answers agree?

   Hint (b):

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

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

   Steps (c):

   also sketch the $f'(x)$ 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 $h$.