Derivatives of Sums

Example:

G = (2 + ln(2)) z⁸ + 2*z - 5

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:

( (2 + ln(2)) z⁸ )

( 2*z )

( 5 )

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,

( (2 + ln(2)) z⁸ )

( 2*z )

( 5 )

we can think of as

f(z)

g(z)

h(z)

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

( (2 + ln(2)) z⁸ )'	=	(0 + 0) z⁸ + 8*(2 + ln(2)) z⁷	(by the product rule, and the derivative rule for sums, and power rule)
( 2*z )'	=	2	(by the rule for constant multiples, and the derivative rule for variables)
( 5 )'	=	0	(by the derivative rule for constants)

so the finished derivative is

G '	=	(0 + 0) z⁸ + 8*(2 + ln(2)) z⁷	+	2	-	0
	=	8*z⁷ (2 + ln(2)) + 2

Page Generated: Sun Mar 1 15:55:05 2026
Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose, University of Michigan Math Dept.