The Product Rule

Let two functions be given:

We know that their derivatives are:

f x x

g x x

( )

( )

2 5

3 1

f xg x

' ( )' ( )

23

If you are given the function written as their product and asked to take the derivative,

your first impulse would probably be to take the product of their derivatives as such:

( ) ( )

(2 5)(3 1)

y f x g x

y x x

' '( ) '( )

' 2 3

' 6

y f x g x

y

y

SO, here is your result:

BUT recall that a constant rate of change should only occur for linear functions.

If we multiply the expressions in “y”, we do

not get a linear function, but a quadratic one. Thus, we recognize that the derivative of a product is NOT the product of the derivatives.

(2 5)(3 1)

' 6

y x x

y

So if this isn’t the right way to find the derivative, what should we do?

One way to find the derivative of their product is to first multiply them together, then differentiate as follows.

2

( ) ( )

(2 5)(3 1)

6 13 5

12 13SO,

y f x g x

x x

x x

dyx

dx

It turns out that this is equal to a sum, where one function is held constant while the derivative of the other is taken and then vice-versa, as follows:

( ) ( )

( ) '( ) ( ) '( )

(2 5)(3) (3 1)(2)

6 15 6 2

12 13

y f x g x

dyf x g x g x f x

dxx x

x x

x

For

We call this the product rule and it can be remembered like this.

Given:

To find the derivative:

Look at the following slide to see this used on our problem.

( ) ( )

( ) ( )

' ( ) '( ) ( ) '( )

' st nd nd st

OR fi rst second

OR (1 )(deriv of 2 ) + (2 )(deriv of 1 )

y f x g x

y

y f x g x g x f x

y

See how this is used on our example:

(2 5)(3 1)

' (2 5) (3) (3 1) (2)

y x x

y x x

first second

first Deriv of second

second Deriv of first

Multiplying . . .

' (2 5)(3) (3 1)(2)

' 6 15 6 2

' 12 13

y x x

y x x

y x

Compare this answer to the answer on slide 6!

Now you may say, “Well then, I’ll just always multiply first.”

1. That’s not always the easiest method, and

2. It won’t work for more complicated functions.

• You try one. Differentiate the following function.

The answer is on the next screen, but don’t look until you attempt the problem on your own.

y x x x x ( )( )3 24 7 3

Use the product rule to find the derivative.

3 2

3 2 2

( 4 )( 7 3)

' ( 4 )(2 7) ( 7 3)(3 4)

y x x x x

y x x x x x x

first second

first Derivative of

second

second Derivative of first

Now, work to UNDERSTAND this application. There are several parts so have patience!

The monthly sales of a new CD-ROM drive are given by

hundred units per month months after being introduced on the market.

Compute and

2( ) 30 0.5q t t t t

(3)

'(3)

q

q

Answers: 1st

2nd

2

2

( ) 30 0.5

(3) 30(3) 0.5(3)

(3) 90 0.5(9)

(3) 90 4.5 85.5

q t t t

q

q

q

Interpretation:

3 months after introduction into the market, 8550 CD-ROMs are being sold per month.

2( ) 30 0.5

'( ) 30

'(3) 30 3

'(3) 27

q t t t

q t t

q

q

Interpretation:

At this same time, the monthly quantity sold is increasing at a rate of 2700 per month.

Continuing: For this same CD-ROM drive, the

retail price in $ is given by , months after

being introduced on the market.

Compute and And interpret.

2( ) 220p t t t

(3)

'(3)

p

p

Answers: 1st

2nd

Interpretation:

3 months after introduction into the market, the CD-ROM drive is being sold for $211.

Interpretation:

At this same time, the retail price is decreasing at a rate of $6 per month.

2

2

( ) 220

(3) 220 3

(3) 220 9

(3) 211

p t t

p

p

p

2( ) 220

'( ) 2

'(3) 2(3)

'(3) 6

p t t

p t t

p

p

And more . . . Revenue for a company selling

a product can be determined by

Revenue = (# of items)(price per item).

Determine the revenue function for the CD-ROM drive described in the past slides.

Recall:

2

2

( ) 30 0.5

( ) 220

q t t t

p t t

Without multiplying: Revenue = (# of items)(price per

item).

Compute and

2 2( ) (30 0.5 )(220 )R t t t t

(3) '(3) R R

First:

2 2( ) (30 0.5 )(220 )

(3) (85.5 )($211)

(3) (8550)($211)

(3) $1,804,050

hundred

R t t t t

R

R

R

Interpretation:

3 months after introduction into the market, revenue is $1,804,050.

Using the product rule and our results from previous evaluations:

2 2

2 2

( ) (30 0.5 )(220 )

'( ) (30 0.5 )( 2 ) (220 )(30 )

'(3) (85.5 )( $6) ($211)(27 )

'(3) $(8550)( 6) (211)(2700)

'(3) $518,400

hundred hundred

R t t t t

R t t t t t t

R

R

R

Interpretation:

3 months after introduction into the market, revenue is increasing at a rate of $518,400 per month.