Derivatives of Sums
Example:
C(y) = (5*y4 - cos(2))7 + y-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:
| C(y) |
= |
( (5*y4 - cos(2))7 ) | + |
( y-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,
| C(y) |
= |
( (5*y4 - cos(2))7 ) | + |
( y-p ) |
we can think of as
So the derivative is
| C '(y) |
= ( |
f(y) |
+ |
g(y) |
)' |
| |
= |
f '(y) |
+ |
g '(y) |
|
| |
= |
( (5*y4 - cos(2))7) ' |
+ |
( y-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*y4 - cos(2))7 )' |
= |
7*(5*y4 - cos(2))6 (5*4*y3 - 0) |
(by the chain rule) |
| ( y-p )' |
= |
(-p)*y-p-1 |
(by the power rule) |
so the finished derivative is
| C '(y) |
= |
7*(5*y4 - cos(2))6 (5*4*y3 - 0) |
+ |
(-p)*y-p-1 |
|
| |
= |
140*y3 (5*y4 - cos(2))6 + (-p)*y-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: Sat Jan 17 17:01:06 2026
Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose,
University of Michigan Math Dept.