Derivatives of Sums
Example:
y = (10*x + 8)e - x-5
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 |
= |
( (10*x + 8)e ) | - |
( x-5 ) |
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 |
= |
( (10*x + 8)e ) | - |
( x-5 ) |
we can think of as
So the derivative is
| y ' |
= ( |
f(x) |
- |
g(x) |
)' |
| |
= |
f '(x) |
- |
g '(x) |
|
| |
= |
( (10*x + 8)e) ' |
- |
( x-5) ' |
|
and we just need to know each of the derivatives on the right-hand
side of the equation. In this case these are
| ( (10*x + 8)e )' |
= |
e*(10*x + 8)e-1 (10 + 0) |
(by the chain rule) |
| ( x-5 )' |
= |
(-5)*x-6 |
(by the power rule) |
so the finished derivative is
| y ' |
= |
e*(10*x + 8)e-1 (10 + 0) |
- |
(-5)*x-6 |
|
| |
= |
(10e)*(10*x + 8)e-1 + 5*x-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 Jun 30 18:10:27 2025
Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose,
University of Michigan Math Dept.