Derivatives of Sums

Example:

y = 6*(sin(p))⁴ - 7

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:

( 6*(sin(p))⁴ )

( 7 )

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,

( 6*(sin(p))⁴ )

( 7 )

we can think of as

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

( 6*(sin(p))⁴ )'	=	64(sin(p))³ cos(p)	(by the rule for constant multiples, and the chain rule)
( 7 )'	=	0	(by the derivative rule for constants)

so the finished derivative is

y '	=	64(sin(p))³ cos(p)	-	0
	=	24*(sin(p))³ cos(p)

Page Generated: Wed Jan 14 14:17:28 2026
Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose, University of Michigan Math Dept.