Derivatives of Sums

Example:

y = (1/4)*t + e^(-1/4)*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:

( (1/4)*t )

( e^(-1/4)*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,

( (1/4)*t )

( e^(-1/4)*t )

we can think of as

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

( (1/4)*t )'	=	1/4	(by the rule for constant multiples, and the derivative rule for variables)
( e^(-1/4)*t )'	=	(-1/4)e^(-1/4)t	(by the chain rule)

so the finished derivative is

y '	=	1/4	+	(-1/4)e^(-1/4)t
	=	1/4 - (1/4)e^(-1/4)t

Page Generated: Sat Apr 19 15:18:00 2025
Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose, University of Michigan Math Dept.