Derivatives of Constant Multiples
Example:
f(x) = (2/3)*cos(sqrt(x) - 2*x)
The first thing to notice when finding the derivative of this function
is that it is
the product of a constant and another function,
as shown in color below:
| f(x) |
= |
2/3 |
cos(sqrt(x) - 2*x) |
The Derivative Rule for Constant Multiples:
The derivative of a constant multiple is the constant times thederivative of the function.
If
then the derivative of
z is
| |
z' |
= |
( c |
f(x) )' |
| |
|
= |
c |
f '(x) |
So our example,
| f(x) |
= |
2/3 |
cos(sqrt(x) - 2*x) |
we can think of as
So the derivative is
| f '(x) |
= ( |
c |
g(x) |
)' |
| |
= |
c |
g '(x) |
|
| |
= |
2/3 |
(cos(sqrt(x) - 2*x))' |
|
and we just need to know the derivative on the right-hand
side of the equation. In this case this is
| cos(sqrt(x) - 2*x) |
= |
(-1)*sin(sqrt(x) - 2*x) ((1/2)*x-1/2 - 2) |
(by the chain rule) |
so the finished derivative is
| f '(x) |
= |
2/3 |
( (-1)*sin(sqrt(x) - 2*x) ((1/2)*x-1/2 - 2) ) |
| |
= |
(-2/3)*sin(sqrt(x) - 2*x) ((1/2)*x-1/2 - 2) |
![[]](/images/boxby.gif)
additional explanation for the derivative of constant multiples
see another derivative of constant multiples example
practice gateway test
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Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose,
University of Michigan Math Dept.