Derivatives of Sums

Example:

R = (sin(1.5))*x + 10*3^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:

( (sin(1.5))*x )

( 10*3^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,

( (sin(1.5))*x )

( 10*3^x )

we can think of as

f(x)

g(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

( (sin(1.5))*x )'	=	(sin(1.5))	(by the rule for constant multiples, and the derivative rule for variables)
( 10*3^x )'	=	10(ln(3))3^x	(by the rule for constant multiples, and the derivative rules for basic functions)

so the finished derivative is

R '	=	(sin(1.5))	+	10(ln(3))3^x
	=	(sin(1.5)) + (10(ln(3)))*3^x

Page Generated: Sat Feb 21 07:07:46 2026
Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose, University of Michigan Math Dept.