CALCULUS Understanding Its Concepts and Methods

Higher derivatives

The second derivative of the function
*f*
is the function
*f '' = *(*f ' *)*'*.
It is the derivative of the derivative.

There are a number of ways to write the second derivative of
*y* = *f*(*x*) .
These
include

The derivative of
*f '' * is denoted
*f ''' * and is called the third derivative of
*f*.
It is also written as
*f *^{(3)}
.
The derivative of
*f *^{(3)}
is
*f *^{(4)}
,
the fourth derivative of
*f*.
In general:

The derivative of the
*n*th
derivative is the
*n*-plus-first
derivative:
*f* ^{(n+1)} =
( *f* ^{(n)} )*'*

The parentheses in the exponent distinguish the
*n*th
derivative
*f* ^{(n)}
from the
*n*th
power
*f* ^{n}
of
*f*.

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Copyright © 2006 Darel Hardy, Fred Richman, Carol Walker, Robert Wisner. All rights reserved. Except upon the express prior permission in writing, from the authors, no part of this work may be reproduced, transcribed, stored electronically, or transmitted in any form by any method.