Derivatives of Constant Multiples

Example:

y = (p)*8^x

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:

8^x

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,

8^x

we can think of as

f(x)

So the derivative is

y '	= (	c	f(x)	)'
	=	c	f '(x)
	=	p	(8^x)'

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

8^x

(ln(8))*8^x

(by the derivative rules for basic functions)

so the finished derivative is

y '	=	p	( (ln(8))*8^x )
	=	(p(ln(8)))*8^x

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.