On-line Math 21

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)
=

lim
h® 0 
1
  ___
Öx+h
 
+Öx
, cancelling out the h top and bottom,
=
1
2Öx
.
Finally, now that the limit is no longer a 0/0 form, you can take the limit as h® 0 .

Copyright (c) 2000 by David L. Johnson.


File translated from TEX by TTH, version 2.61.
On 24 Oct 2000, 22:42.