§2.2 Product and Quotient Rules

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§2.2 Product and Quotient Rules

The student will learn about derivatives

marginal averages as used in business and economics, and

involving products, involving products, quotients,

higher order derivatives.

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Practice – find derivatives for:

1. y = 3x 4 + x 3 – 2 x 2 + 7x - 5

y’ = - 12x - 5 - 3x - 4 + 4 x - 3 - 7x - 2

3. y = 3x 7/4 + x 2/3 – 2 x – 1/2 + 7x -11/5 - 5

y’ = 12x 3 + 3x 2 – 4 x + 7

2. y = 3x - 4 + x - 3 – 2 x - 2 + 7x - 1 - 5

516233143 x5

77xx32x

421'y

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Practice – find derivatives for:

4. 3 xy 31x

6. Find the equation of the line tangent to

y = x 2 – 4x + 5 at x = 3.

5.5x

7y

32x31'y

25x7 27x2

35'y

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Derivates of ProductsThe derivative of the product of two functions is the first function times the derivative of the second function plus the second function times the derivative of the first function.

Product Rule

)x('f)x(s)x('s)x(f])x(s)x(f[dxd

OR 'fs'sf)sf(dxd

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Important NoteThe derivative of the product is NOT the product of the derivatives.

's'f)sf(dxd

'fs'sf)sf(dxd

It is

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ExampleFind the derivative of y = 5x2(x3 + 2).

Product Rule

Let f (x) = 5x2 then f ‘ (x) =Let s (x) = x3 + 2 then s ‘ (x) =

= 15x4 + 10x4 + 20x = 25x4 + 20x

10x3x2, and

)]x('f)x(s)x('s)x(f])x(s)x(f[dxd

y ‘ (x) = 5x2 · 3x2 + (x3 + 2)y ‘ (x) = 5x2y ‘ (x) = 5x2 · 3x2y ‘ (x) = 5x2 · 3x2 + (x3 + 2) · 10x

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Derivatives of Quotients The derivative of the quotient of two functions is the bottom function times the derivative of the top function minus the top function times the derivative of the bottom function, all over the bottom function squared.

Quotient Rule:

2])x(b[)x('b)x(t)x('t)x(b

)x(b)x(t

dxd

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Derivatives of Quotients

May also be expressed as -

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d t (x) b t ' t b'dx b (x) b

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Example

Let t (x) = 3x and then t ‘ (x) =

Find the derivative of .5x2

x3y

Let b (x) = 2x + 5 and then b ‘ (x) =

2)5x2(

2x33)5x2()x('f 2)5x2(15

3.2.

2

d t (x) b t ' t b'dx b (x) b

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Marginal Average CostIf x is the number of units of a product produced in some time interval, then

Average cost per unit = x

)x(C)x(C

Marginal average cost = )x(Cdxd)x('C

This describes how the average cost changes if you produce one more item!

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Marginal Average RevenueIf x is the number of units of a product sold in some time interval, then

Average revenue per unit = x

)x(R)x(R

Marginal average revenue = )x(Rdxd)x('R

This describes how the average revenue changes if you produce one more item!

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Marginal Average Profit. If x is the number of units of a product produced and sold in some time interval, then

Average profit per unit = x

)x(P)x(P

Marginal average profit = )x(Pdxd)x('P

This describes how the average profit changes if you produce one more item!

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Marginal AveragesIf C (x) is a function that describes how the total cost is calculated,Then the marginal cost is the cost of the next unit produced (the rate of change in the cost), and

the average cost is the total cost divided by the number of units produced, and

the marginal average cost is the change in the average cost if you produce one more unit.

The above is also true for revenue and profit.

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Warning!To calculate the marginal averages you must calculate the average first (divide by x) and then the derivative. If you change this order you will get no useful economic interpretations.

STOP

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Example 2The total cost of printing x dictionaries is

C (x) = 20,000 + 10x1. Find the average cost per unit if 1,000 dictionaries are produced.

= $30

x

)x(C)x(C

)1000(C1000

000,10000,20

xx1020000

What does this mean?

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Example 2 continuedThe total cost of printing x dictionaries is

C (x) = 20,000 + 10x 2. Find the marginal average cost at a production level of 1,000 dictionaries, and interpret the results.


x

x1020000dxd)x('C

2100020000)1000('C

2x20000

02.0What does this mean?

2x20000

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Example 2 concludedThe total cost of printing x dictionaries is

C (x) = 20,000 + 10x 3. Use the results from above to estimate the average cost per dictionary if 1,001 dictionaries are produced. Average cost = $30.00Marginal average cost = - 0.02 The average cost per dictionary for 1001 dictionaries would be the average for 1000 plus the marginal average cost, or

$30.00 + (- 0.02) = $29.98

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The Second DerivativeThe derivative of the derivative is called the second derivative and has some useful applications.

Notation -

2

2

2

2

2

dxyd

)x(fdxd

y"y

)x("f

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Other Higher-Order DerivativesYou may continue to take derivatives of derivatives.

Notation -

.etc)x(f)x(f)x('''f

)x("f)x('f

)5(

)4(

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ExampleFind the first four derivatives of

y = x 3 + 4 x 2 - 7 x + 5

f ′ (x) =

f ′′ (x) =

f ′′′ (x) = 6

f (4) =

3 x 2 + 8 x - 7

6x + 8

0

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Higher-Order DerivativesHigher-order derivatives sometimes involve the product or the quotient rules. Take your time and organize your work and you should do fine.

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Distance, Velocity and Acceleration(A First Application)

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Example

1. Find its distance when t = 4.

After t hours a train is s(t) = 24 t 2 – 2 t 3 miles from its starting point.

s (4) = 24 · 4 2 – 2 · 4 3 = 384 – 128 = 256 miles

2. Find its velocity when t = 4.

s’ (t) = 48 t – 6 t 2 ands’ (4) = 48 · 4 – 6 · 4 2 = 192 – 96 = 96 mph

Use your calculator.

Use your calculator.



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Example Continued

1. Find its distance when t = 4. [256 miles]

After t hours a train is s (t) = 24 t 2 – 2 t 3 miles from its starting point.

2. Find its velocity when t = 4. [96 mph]

s” (t) = 48 – 12 t and

s” (4) = 48– 12 · 4 = 48 – 48 = 0

3. Find its acceleration when t = 4.

s’ (t) = 48 t – 6 t 2 and


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Summary.

Product Rule. If f (x) and s (x), then

f • s ' + s • f ' sfdxd

Quotient Rule. If t (x) and b (x), then

2b'bt'tb

bt

dxd

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Summary.


Marginal average revenue = )x(Rdxd)x('R

Marginal average profit = )x(Pdxd)x('P

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Summary.

We learned about higher-order derivatives. That is, derivatives of derivatives.

We saw one application of the second derivative.

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ASSIGNMENT

§2.2 on my website

13, 14, 15, 16, 17, 18, 26, 27.

§2.2 Product and Quotient Rules

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