Derivatives of Sums

Example:

z = 3*y⁶ + 6*y + e

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:

( 3*y⁶ )

( 6*y )

( e )

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,

( 3*y⁶ )

( 6*y )

( e )

we can think of as

f(y)

g(y)

h(y)

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

( 3*y⁶ )'	=	36y⁵	(by the rule for constant multiples, and the power rule)
( 6*y )'	=	6	(by the rule for constant multiples, and the derivative rule for variables)
( e )'	=	0	(by the derivative rule for constants)

so the finished derivative is

z '	=	36y⁵	+	6	+	0
	=	18*y⁵ + 6

Page Generated: Sun Dec 28 19:10:26 2025
Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose, University of Michigan Math Dept.