Derivatives of Sums

Example:

L(p) = (cos(2))*p - (p)*2^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:

L(p)

( (cos(2))*p )

( (p)*2^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,

L(p)

( (cos(2))*p )

( (p)*2^p )

we can think of as

L(p)

f(p)

g(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

( (cos(2))*p )'	=	(cos(2))	(by the rule for constant multiples, and the derivative rule for variables)
( (p)*2^p )'	=	(p)(ln(2))2^p	(by the rule for constant multiples, and the derivative rules for basic functions)

so the finished derivative is

L '(p)	=	(cos(2))	-	(p)(ln(2))2^p
	=	(cos(2)) - (p(ln(2)))*2^p

Page Generated: Mon Mar 2 04:15:23 2026
Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose, University of Michigan Math Dept.