Derivatives of Sums

Example:

C(x) = 3*x - e*(3)/(2)^x

The first thing to notice when finding the derivative of this function is that it is the sum of several terms, as shown in color below:

C(x)

( 3*x )

( e*(3)/(2)^x )

The Derivative Rule for Sums:

The derivative of a sum is the sum of the derivatives.
If

( f(x)

g(x) )

then the derivative of z is

	z'	=	( f(x)	+	g(x) )'
		=	f '(x)	+	g'(x)

So our example,

C(x)

( 3*x )

( e*(3)/(2)^x )

we can think of as

C(x)

f(x)

g(x)

So the derivative is

and we just need to know each of the derivatives on the right-hand side of the equation. In this case these are

( 3*x )'	=	3	(by the rule for constant multiples, and the derivative rule for variables)
( e*(3)/(2)^x )'	=	e(ln((3)/(2)))(3)/(2)^x	(by the rule for constant multiples, and the derivative rules for basic functions)

so the finished derivative is

C '(x)	=	3	-	e(ln((3)/(2)))(3)/(2)^x
	=	3 - (e((ln((3)/(2)))))*(3)/(2)^x

Page Generated: Wed Dec 10 06:30:03 2025
Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose, University of Michigan Math Dept.