Derivatives of Constant Multiples
Example:
W = 6*ln(sqrt(t))
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
| W ' |
= ( |
c |
f(t) |
)' |
| |
= |
c |
f '(t) |
|
| |
= |
6 |
(ln(sqrt(t)))' |
|
and we just need to know the derivative on the right-hand
side of the equation. In this case this is
| ln(sqrt(t)) |
= |
(1/2)*(sqrt(t))-1 t-1/2 |
(by the chain rule) |
so the finished derivative is
| W ' |
= |
6 |
( (1/2)*(sqrt(t))-1 t-1/2 ) |
| |
= |
3*t-1/2 (sqrt(t))-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: Thu May 7 16:57:03 2026
Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose,
University of Michigan Math Dept.