On-line Math 21

2.1  The definition of the derivative

Example 3 Find f¢(2) for the function f(x) = x³ .

Solution

This is similar to the example of the slope of a curve .

f¢(2) = lim h® 0  f(2+h)-f(2) h = lim h® 0  (2+h)3-(2)3 h = lim h® 0  23+3·22h+3·2h2+h3-23 h = lim h® 0  12h+6h2+h3 h = lim h® 0  12+6h+h2 = 12.

Once you cancel the 2³ 's, then it is clear that you can cancel out a factor of h top and bottom, and after that, the limit is simple.

Copyright (c) 2000 by David L. Johnson.