Derivatives of Sums

Example:

h(x) = e^(-1)*x + e^6*x

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(x)

( e^(-1)*x )

( e^6*x )

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(x)

( e^(-1)*x )

( e^6*x )

we can think of as

h(x)

f(x)

g(x)

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^(-1)*x )'	=	(-1)e^(-1)x	(by the chain rule)
( e^6*x )'	=	6e^6x	(by the chain rule)

so the finished derivative is

h '(x)	=	(-1)e^(-1)x	+	6e^6x
	=	(-1)e^(-1)x + 6e^6x

Page Generated: Sun Dec 28 10:32:49 2025
Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose, University of Michigan Math Dept.

h '(x)	=	(-1)e^(-1)x	+	6e^6x
	=	(-1)e^(-1)x + 6e^6x