Derivatives of Sums
Example:
y = (6*p4 + 10)7 + p-6
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:
| y |
= |
( (6*p4 + 10)7 ) | + |
( p-6 ) |
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,
| y |
= |
( (6*p4 + 10)7 ) | + |
( p-6 ) |
we can think of as
So the derivative is
| y ' |
= ( |
f(p) |
+ |
g(p) |
)' |
| |
= |
f '(p) |
+ |
g '(p) |
|
| |
= |
( (6*p4 + 10)7) ' |
+ |
( p-6) ' |
|
and we just need to know each of the derivatives on the right-hand
side of the equation. In this case these are
| ( (6*p4 + 10)7 )' |
= |
7*(6*p4 + 10)6 (6*4*p3 + 0) |
(by the chain rule) |
| ( p-6 )' |
= |
(-6)*p-7 |
(by the power rule) |
so the finished derivative is
| y ' |
= |
7*(6*p4 + 10)6 (6*4*p3 + 0) |
+ |
(-6)*p-7 |
|
| |
= |
168*p3 (6*p4 + 10)6 - 6*p-7 |
![[]](/images/boxby.gif)
additional explanation for the derivative of sums
see another derivative of sums example
practice gateway test
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Page Generated: Sun Jan 18 09:29:03 2026
Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose,
University of Michigan Math Dept.