Derivatives of Products

Example:

y = cos(2*x + 5) ((1/4)*x + cos(5))

The first thing to notice when finding the derivative of this function is that it is the product of several terms, as shown in color below:

( cos(2*x + 5))

( (1/4)*x + cos(5))

The Derivative Rule for Products:

The derivative of a product is the derivative of the first term times the second (and third, etc.) term(s), plus the first (and third, etc.) term(s) times the derivative of the second, etc.
If

(f(x) g(x))

then the derivative of z is

	z '	=	(f(x) g(x))'
		=	f '(x) g(x)	+	f(x) g '(x)

So our example,

( cos(2*x + 5))

( (1/4)*x + cos(5))

we can think of as

f(x)

g(x)

So the derivative is

y '	= (	f(x)	g(x)	)'
	=	f '(x)	g(x)	+	f(x)	g '(x)
	=	( cos(2*x + 5) )'	( (1/4)*x + cos(5) )	+	( cos(2*x + 5) )	( (1/4)*x + cos(5) )'

and we just need to know each of the derivatives on the right-hand side of the equation. In this case these are

( cos(2*x + 5) )'	=	(-1)sin(2x + 5) (2 + 0)	(by the chain rule)
( (1/4)*x + cos(5) )'	=	1/4 + 0	(by the derivative rule for sums, rule for constant multiples, and the derivative rule for constants)

so the finished derivative is

y '	=	( (-1)sin(2x + 5) (2 + 0) )	( (1/4)*x + cos(5) )	+	( cos(2*x + 5) )	( 1/4 + 0 )
	=	(-2)sin(2x + 5) ((1/4)x + cos(5)) + (1/4)cos(2*x + 5)

additional explanation for the product rule
see another product rule example
practice gateway test
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Page Generated: Sun Dec 21 00:44:03 2025
Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose, University of Michigan Math Dept.