Derivatives of Sums

Example:

W = (e)^-4 - e^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:

( (e)^-4 )

( e^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,

( (e)^-4 )

( e^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

( (e)^-4 )'	=	0	(by the derivative rule for constants)
( e^4*t )'	=	4e^4t	(by the chain rule)

so the finished derivative is

W '	=	0	-	4e^4t
	=	(-4)e^4t

Page Generated: Wed Jan 21 08:56:55 2026
Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose, University of Michigan Math Dept.