Derivatives of Sums
Example:
R = (5*s3 - 3)p - s-p
The first thing to notice when finding the derivative of this function
is that it is
the sum of several terms,
as shown in color below:
| R |
= |
( (5*s3 - 3)p ) | - |
( s-p ) |
The Derivative Rule for Sums:
The derivative of a sum is the sum of the derivatives.
If
then the derivative of
z is
| |
z' |
= |
( f(x) |
+ |
g(x) )' |
| |
|
= |
f '(x) |
+ |
g'(x) |
So our example,
| R |
= |
( (5*s3 - 3)p ) | - |
( s-p ) |
we can think of as
So the derivative is
| R ' |
= ( |
f(s) |
- |
g(s) |
)' |
| |
= |
f '(s) |
- |
g '(s) |
|
| |
= |
( (5*s3 - 3)p) ' |
- |
( s-p) ' |
|
and we just need to know each of the derivatives on the right-hand
side of the equation. In this case these are
| ( (5*s3 - 3)p )' |
= |
(p)*(5*s3 - 3)p-1 (5*3*s2 - 0) |
(by the chain rule) |
| ( s-p )' |
= |
(-p)*s-p-1 |
(by the power rule) |
so the finished derivative is
| R ' |
= |
(p)*(5*s3 - 3)p-1 (5*3*s2 - 0) |
- |
(-p)*s-p-1 |
|
| |
= |
(15p)*s2 (5*s3 - 3)p-1 - (-p)*s-p-1 |
![[]](/images/boxby.gif)
additional explanation for the derivative of sums
see another derivative of sums example
practice gateway test
previous page
Page Generated: Mon Jan 12 19:57:54 2026
Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose,
University of Michigan Math Dept.