Derivatives of Constant Multiples

Example:

y = a*(x - b) (x - c) (x - d)

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:

(x - b) (x - c) (x - d)

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,

(x - b) (x - c) (x - d)

we can think of as

f(x)

So the derivative is

y '	= (	c	f(x)	)'
	=	c	f '(x)
	=	a	((x - b) (x - c) (x - d))'

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

(x - b) (x - c) (x - d)

(1 - 0) (x - c) (x - d) + (x - b) (1 - 0) (x - d) + (x - b) (x - c) (1 - 0)

(by the product rule, and the derivative rule for sums, derivative rule for sums (again), and derivative rule for sums (again))

so the finished derivative is

y '	=	a	( (1 - 0) (x - c) (x - d) + (x - b) (1 - 0) (x - d) + (x - b) (x - c) (1 - 0) )
	=	a*((x - c) (x - d) + (x - b) (x - d) + (x - b) (x - c))

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.