Derivatives of Constant Multiples
Example:
g(z) = (e3)*sin(5*z2 - 7*z - 6)
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:
| g(z) |
= |
(e3) |
sin(5*z2 - 7*z - 6) |
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,
| g(z) |
= |
(e3) |
sin(5*z2 - 7*z - 6) |
we can think of as
So the derivative is
| g '(z) |
= ( |
c |
f(z) |
)' |
| |
= |
c |
f '(z) |
|
| |
= |
(e3) |
(sin(5*z2 - 7*z - 6))' |
|
and we just need to know the derivative on the right-hand
side of the equation. In this case this is
| sin(5*z2 - 7*z - 6) |
= |
cos(5*z2 - 7*z - 6) (5*2*z - 7 - 0) |
(by the chain rule) |
so the finished derivative is
| g '(z) |
= |
(e3) |
( cos(5*z2 - 7*z - 6) (5*2*z - 7 - 0) ) |
| |
= |
(e3)*cos(5*z2 - 7*z - 6) (10*z - 7) |
![[]](/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.