Derivatives of Constant Multiples

Example:

y = (-10)*x^e

The first thing to notice when finding the derivative of this function is that it is the product of a constant and another function, as shown in color below:

-10

x^e

The Derivative Rule for Constant Multiples:

The derivative of a constant multiple is the constant times thederivative of the function.
If

( f(x) )

then the derivative of z is

	z'	=	( c	f(x) )'
		=	c	f '(x)

So our example,

-10

x^e

we can think of as

f(x)

So the derivative is

y '	= (	c	f(x)	)'
	=	c	f '(x)
	=	-10	(x^e)'

and we just need to know the derivative on the right-hand side of the equation. In this case this is

x^e

e*x^e-1

(by the power rule)

so the finished derivative is

y '	=	-10	( e*x^e-1 )
	=	(-10e)*x^e-1

additional explanation for the derivative of constant multiples
see another derivative of constant multiples example
practice gateway test
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Page Generated: Mon Apr 27 09:02:32 2026
Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose, University of Michigan Math Dept.