Derivatives of Constant Multiples

Example:

f(t) = 24*1.04^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:

f(t)

1.04^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,

f(t)

1.04^t

we can think of as

f(t)

g(t)

So the derivative is

f '(t)	= (	c	g(t)	)'
	=	c	g '(t)
	=	24	(1.04^t)'

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

1.04^t

(ln(1.04))*1.04^t

(by the derivative rules for basic functions)

so the finished derivative is

f '(t)	=	24	( (ln(1.04))*1.04^t )
	=	(24(ln(1.04)))*1.04^t

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.