On-line Math 21

2.1  The definition of the derivative

Example 4 Find f¢(x) for the function f(x) = Öx .

Solution

f¢(x) = lim h® 0  f(x+h)-f(x) h = lim h® 0    ___ Öx+h   -Öx h = lim h® 0  æ è   ___ Öx+h   -Öx ö ø æ è   ___ Öx+h   +Öx ö ø h æ è   ___ Öx+h   +Öx ö ø = lim h® 0  (x+h)-x h(   ___ Öx+h   +Öx) ,

multiplying out the numerator. Remember,

(a-b)(a+b) = a2-b2.

Note, now, that the numerator is just h . Cancel that out, top and bottom, which will mean that the limit is no longer an indeterminate form of type 0/0 .

Finish it

Copyright (c) 2000 by David L. Johnson.