Derivatives of Constant Multiples
Example:
y = (p)*ln(5*x2 + p)
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
| y ' |
= ( |
c |
f(x) |
)' |
| |
= |
c |
f '(x) |
|
| |
= |
p |
(ln(5*x2 + p))' |
|
and we just need to know the derivative on the right-hand
side of the equation. In this case this is
| ln(5*x2 + p) |
= |
(5*x2 + p)-1 (5*2*x + 0) |
(by the chain rule) |
so the finished derivative is
| y ' |
= |
p |
( (5*x2 + p)-1 (5*2*x + 0) ) |
| |
= |
(10p)*x (5*x2 + p)-1 |
![[]](/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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Page Generated: Sun Jan 18 10:52:02 2026
Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose,
University of Michigan Math Dept.