Derivatives of Sums
Example:
C(q) = (-4)*(3*q4 + 3)4 - 6*q - 3
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:
| C(q) |
= |
( (-4)*(3*q4 + 3)4 ) | - |
( 6*q ) |
- |
( 3 ) |
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,
| C(q) |
= |
( (-4)*(3*q4 + 3)4 ) | - |
( 6*q ) |
- |
( 3 ) |
we can think of as
| C(q) |
= |
f(q) |
- |
g(q) |
- |
h(q) |
So the derivative is
| C '(q) |
= ( |
f(q) |
- |
g(q) |
- |
h(q) |
)' |
| |
= |
f '(q) |
- |
g '(q) |
- |
h '(q) |
|
| |
= |
( (-4)*(3*q4 + 3)4) ' |
- |
( 6*q) ' |
- |
( 3) ' |
|
and we just need to know each of the derivatives on the right-hand
side of the equation. In this case these are
so the finished derivative is
| C '(q) |
= |
(-4)*4*(3*q4 + 3)3 (3*4*q3 + 0) |
- |
6 |
- |
0 |
|
| |
= |
(-192)*q3 (3*q4 + 3)3 - 6 |
![[]](/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: Mon Dec 15 09:47:30 2025
Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose,
University of Michigan Math Dept.