Derivatives of Constant Multiples
Example:
f(z) = 7*ln(5*z-5)
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:
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,
we can think of as
So the derivative is
| f '(z) |
= ( |
c |
g(z) |
)' |
| |
= |
c |
g '(z) |
|
| |
= |
7 |
(ln(5*z-5))' |
|
and we just need to know the derivative on the right-hand
side of the equation. In this case this is
| ln(5*z-5) |
= |
5*(5*z-5)-1 (-5)*z-6 |
(by the chain rule) |
so the finished derivative is
| f '(z) |
= |
7 |
( 5*(5*z-5)-1 (-5)*z-6 ) |
| |
= |
(-175)*z-6 (5*z-5)-1 |
![[]](/images/boxby.gif)
additional explanation for the derivative of constant multiples
see another derivative of constant multiples example
practice gateway test
previous page
Page Generated: Wed Dec 17 04:01:27 2025
Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose,
University of Michigan Math Dept.