Derivatives of Constant Multiples

Example:

y = 2*r e^3*r

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:

r e^3*r

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,

r e^3*r

we can think of as

f(r)

So the derivative is

y '	= (	c	f(r)	)'
	=	c	f '(r)
	=	2	(r e^3*r)'

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

r e^3*r

e^3*r + 3*r e^3*r

(by the product rule, and the derivative rule for variables, and chain rule)

so the finished derivative is

y '	=	2	( e^3r + 3r e^3*r )
	=	2(e^3r + 3r e^3r)

additional explanation for the derivative of constant multiples
see another derivative of constant multiples example
practice gateway test
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Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose, University of Michigan Math Dept.