Derivatives of Sums
Example:
r(t) = sqrt(3*t2 - 4) - t7 + ln(2)
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(t) |
= |
( sqrt(3*t2 - 4) ) | - |
( t7 ) |
+ |
( ln(2) ) |
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(t) |
= |
( sqrt(3*t2 - 4) ) | - |
( t7 ) |
+ |
( ln(2) ) |
we can think of as
| r(t) |
= |
f(t) |
- |
g(t) |
+ |
h(t) |
So the derivative is
| r '(t) |
= ( |
f(t) |
- |
g(t) |
+ |
h(t) |
)' |
| |
= |
f '(t) |
- |
g '(t) |
+ |
h '(t) |
|
| |
= |
( sqrt(3*t2 - 4)) ' |
- |
( t7) ' |
+ |
( ln(2)) ' |
|
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
| r '(t) |
= |
(1/2)*(3*t2 - 4)-1/2 (3*2*t - 0) |
- |
7*t6 |
+ |
0 |
|
| |
= |
3*t (3*t2 - 4)-1/2 - 7*t6 |
![[]](/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: Fri Dec 26 02:28:39 2025
Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose,
University of Michigan Math Dept.