Derivatives of Sums

Example:

h(x) = 8*(sin(x))³ + 1/2

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:

h(x)

( 8*(sin(x))³ )

( 1/2 )

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,

h(x)

( 8*(sin(x))³ )

( 1/2 )

we can think of as

h(x)

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

( 8*(sin(x))³ )'	=	83(sin(x))² cos(x)	(by the rule for constant multiples, and the chain rule)
( 1/2 )'	=	0	(by the derivative rule for constants)

so the finished derivative is

h '(x)	=	83(sin(x))² cos(x)	+	0
	=	24*(sin(x))² cos(x)

Page Generated: Mon Feb 9 11:49:36 2026
Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose, University of Michigan Math Dept.