Derivatives of Sums

Example:

h(t) = t^-(1/2) + 4*sin(sqrt(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)

( t^-(1/2) )

( 4*sin(sqrt(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)

( t^-(1/2) )

( 4*sin(sqrt(t)) )

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

( t^-(1/2) )'	=	(-(1/2))*t^-(1/2)-1	(by the power rule)
( 4*sin(sqrt(t)) )'	=	4(1/2)cos(sqrt(t)) t^-1/2	(by the rule for constant multiples, and the chain rule)

so the finished derivative is

h '(t)	=	(-(1/2))*t^-(1/2)-1	+	4(1/2)cos(sqrt(t)) t^-1/2
	=	(-(1/2))t^-(1/2)-1 + 2t^-1/2 cos(sqrt(t))

Page Generated: Sat Jan 17 00:12:01 2026
Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose, University of Michigan Math Dept.