Derivatives of Constant Multiples

Example:

T(t) = (-a)*e^(-b)*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:

T(t)

-a

e^(-b)*t

The Derivative Rule for Constant Multiples:

The derivative of a constant multiple is the constant times thederivative of the function.
If

( f(x) )

then the derivative of z is

	z'	=	( c	f(x) )'
		=	c	f '(x)

So our example,

T(t)

-a

e^(-b)*t

we can think of as

T(t)

f(t)

So the derivative is

T '(t)	= (	c	f(t)	)'
	=	c	f '(t)
	=	-a	(e^(-b)*t)'

and we just need to know the derivative on the right-hand side of the equation. In this case this is

e^(-b)*t

(-b)*e^(-b)*t

(by the chain rule)

so the finished derivative is

T '(t)	=	-a	( (-b)e^(-b)t )
	=	((-a)(-b))e^(-b)t

additional explanation for the derivative of constant multiples
see another derivative of constant multiples example
practice gateway test
previous page

Page Generated: Fri May 1 08:31:32 2026
Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose, University of Michigan Math Dept.