Derivatives of Sums

Example:

z = ((p)*p)⁵ - 2*p² - p

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:

( ((p)*p)⁵ )

( 2*p² )

( p )

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,

( ((p)*p)⁵ )

( 2*p² )

( p )

we can think of as

f(p)

g(p)

h(p)

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

( ((p)*p)⁵ )'	=	(p)5((p)*p)⁴	(by the chain rule)
( 2*p² )'	=	22p	(by the rule for constant multiples, and the power rule)
( p )'	=	1	(by the derivative rule for variables)

so the finished derivative is

z '	=	(p)5((p)*p)⁴	-	22p	-	1
	=	(5p)((p)p)⁴ - 4*p - 1

Page Generated: Sun Nov 23 06:35:29 2025
Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose, University of Michigan Math Dept.

z '	=	(p)5((p)*p)⁴	-	22p	-	1
	=	(5p)((p)p)⁴ - 4*p - 1