Derivatives of Sums

Example:

z = (1/4)*x - 5*((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:

( (1/4)*x )

( 5*((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,

( (1/4)*x )

( 5*((3)/(2))^x )

we can think of as

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

( (1/4)*x )'	=	1/4	(by the rule for constant multiples, and the derivative rule for variables)
( 5*((3)/(2))^x )'	=	5(ln(((3)/(2))))((3)/(2))^x	(by the rule for constant multiples, and the derivative rules for basic functions)

so the finished derivative is

z '	=	1/4	-	5(ln(((3)/(2))))((3)/(2))^x
	=	1/4 - (5((ln(((3)/(2))))))*((3)/(2))^x

Page Generated: Thu Dec 25 01:24:19 2025
Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose, University of Michigan Math Dept.