Derivatives of Sums
Example:
C = ((sqrt(3))*z + 5)3 + (sqrt(3))*z2 + 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:
| C |
= |
( ((sqrt(3))*z + 5)3 ) | + |
( (sqrt(3))*z2 ) |
+ |
( 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,
| C |
= |
( ((sqrt(3))*z + 5)3 ) | + |
( (sqrt(3))*z2 ) |
+ |
( 5 ) |
we can think of as
So the derivative is
| C ' |
= ( |
f(z) |
+ |
g(z) |
+ |
h(z) |
)' |
| |
= |
f '(z) |
+ |
g '(z) |
+ |
h '(z) |
|
| |
= |
( ((sqrt(3))*z + 5)3) ' |
+ |
( (sqrt(3))*z2) ' |
+ |
( 5) ' |
|
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 ' |
= |
3*((sqrt(3))*z + 5)2 ((sqrt(3)) + 0) |
+ |
(sqrt(3))*2*z |
+ |
0 |
|
| |
= |
(3(sqrt(3)))*((sqrt(3))*z + 5)2 + (2(sqrt(3)))*z |
additional explanation for the derivative of sums
see another derivative of sums example
practice gateway test
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Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose,
University of Michigan Math Dept.