Derivatives of Sums

Example:

h(t) = 7*t⁴ + 5 - 6*t

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(t)

( 7*t⁴ )

( 5 )

( 6*t )

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(t)

( 7*t⁴ )

( 5 )

( 6*t )

we can think of as

h(t)

f(t)

g(t)

p(t)

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

( 7*t⁴ )'	=	74t³	(by the rule for constant multiples, and the power rule)
( 5 )'	=	0	(by the derivative rule for constants)
( 6*t )'	=	6	(by the rule for constant multiples, and the derivative rule for variables)

so the finished derivative is

h '(t)	=	74t³	+	0	-	6
	=	28*t³ - 6

Page Generated: Tue Jan 13 03:11:50 2026
Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose, University of Michigan Math Dept.