Derivatives of Sums

Example:

y = x³ + x² - 4*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:

( x³ )

( x² )

( 4*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,

( x³ )

( x² )

( 4*x )

we can think of as

f(x)

g(x)

h(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

( x³ )'	=	3*x²	(by the power rule)
( x² )'	=	2*x	(by the power rule)
( 4*x )'	=	4	(by the rule for constant multiples, and the derivative rule for variables)

so the finished derivative is

y '	=	3*x²	+	2*x	-	4
	=	3x² + 2x - 4

Page Generated: Thu Dec 4 01:22:33 2025
Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose, University of Michigan Math Dept.