Derivatives of Sums

Example:

V = 10*(ln(q))² - 2

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:

( 10*(ln(q))² )

( 2 )

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,

( 10*(ln(q))² )

( 2 )

we can think of as

f(q)

g(q)

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

( 10*(ln(q))² )'	=	102ln(q) q^-1	(by the rule for constant multiples, and the chain rule)
( 2 )'	=	0	(by the derivative rule for constants)

so the finished derivative is

V '	=	102ln(q) q^-1	-	0
	=	20*q^-1 ln(q)

Page Generated: Sun Feb 8 15:01:35 2026
Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose, University of Michigan Math Dept.