On-line Math 21

2.1  The definition of the derivative

Example 5 Find the derivative of the function f(x) = 2^x .

Solution

At any point x ,

f¢(x) = lim h® 0  2(x+h)-2x h , by the definition, = lim h® 0  2x2h-2x h , since 2(x+h) = 2x2h, = lim h® 0  2x æ ç è 2h-1 h ö ÷ ø , factoring out 2x, = 2x lim h® 0  æ ç è 2h-1 h ö ÷ ø ,

so, in order to know this slope, all we need to do is find that last limit, which is the slope of the curve

f(x) = 2x

at x = 0 . If we're careful in computing it, we'll see that the slope is about 0.69 .

Copyright (c) 2000 by David L. Johnson.