Derivatives of Sums

Example:

h(t) = sqrt(t³) - 2*t⁴

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)

( sqrt(t³) )

( 2*t⁴ )

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)

( sqrt(t³) )

( 2*t⁴ )

we can think of as

h(t)

f(t)

g(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

( sqrt(t³) )'	=	3(1/2)(t³)^-1/2 t²	(by the chain rule)
( 2*t⁴ )'	=	24t³	(by the rule for constant multiples, and the power rule)

so the finished derivative is

h '(t)	=	3(1/2)(t³)^-1/2 t²	-	24t³
	=	(3/2)t^-3/2+2 - 8t³

Page Generated: Fri Nov 28 22:14:32 2025
Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose, University of Michigan Math Dept.

h '(t)	=	3(1/2)(t³)^-1/2 t²	-	24t³
	=	(3/2)t^-3/2+2 - 8t³