1 §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.

2 Practice – find derivatives for: 1. y = 3x 4 + x 3 – 2 x 2 + 7x - 5 y’ = - 12x x x x y = 3x 7/4 + x 2/3 – 2 x – 1/2 + 7x -11/5 - 5 y’ = 12x 3 + 3x 2 – 4 x y = 3x x - 3 – 2 x x

3 Practice – find derivatives for: 4. 6.Find the equation of the line tangent to y = x 2 – 4x + 5 at x = 3. 5.

4 Derivates of Products The 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 OR

5 Important Note The derivative of the product is NOT the product of the derivatives. It is

6 Example Find the derivative of y = 5x 2 (x 3 + 2). Product Rule Let f (x) = 5x 2 then f ‘ (x) = Let s (x) = x then s ‘ (x) = = 15x x x =25x x 10x 3x 2, and y ‘ (x) = 5x 2 · 3x 2 + (x 3 + 2) y ‘ (x) = 5x 2 y ‘ (x) = 5x 2 · 3x 2 y ‘ (x) = 5x 2 · 3x 2 + (x 3 + 2) · 10x

7 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:

8 Derivatives of Quotients May also be expressed as -

9 Example Let t (x) = 3x and then t ‘ (x) = Find the derivative of. Let b (x) = 2x + 5 and then b ‘ (x) = 3. 2.

10 Marginal Average Cost If x is the number of units of a product produced in some time interval, then Average cost per unit = Marginal average cost = This describes how the average cost changes if you produce one more item!

11 Marginal Average Revenue If x is the number of units of a product sold in some time interval, then Average revenue per unit = Marginal average revenue = This describes how the average revenue changes if you produce one more item!

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

13 Marginal Averages If 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.

14 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

15 Example 2 The total cost of printing x dictionaries is C (x) = 20, x 1. Find the average cost per unit if 1,000 dictionaries are produced. = $30 What does this mean?

16 Example 2 continued The total cost of printing x dictionaries is C (x) = 20, x 2. Find the marginal average cost at a production level of 1,000 dictionaries, and interpret the results. Marginal average cost = What does this mean?

17 Example 2 concluded The total cost of printing x dictionaries is C (x) = 20, x 3. Use the results from above to estimate the average cost per dictionary if 1,001 dictionaries are produced. Average cost = $30.00 Marginal average cost = The average cost per dictionary for 1001 dictionaries would be the average for 1000 plus the marginal average cost, or $ (- 0.02) =$29.98

18 The Second Derivative The derivative of the derivative is called the second derivative and has some useful applications. Notation -

19 Other Higher-Order Derivatives You may continue to take derivatives of derivatives. Notation -

20 Example Find the first four derivatives of y = x x x + 5 f ′ (x) = f ′′ (x) = f ′′′ (x) =6 f (4) = 3 x x - 7 6x + 8 0

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

22 Distance, Velocity and Acceleration (A First Application)

23 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 and s’ (4) = 48 · 4 – 6 · 4 2 = 192 – 96 = 96 mph Use your calculator. What does this mean?

24 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 What does this mean?

25 Summary. Product Rule. If f (x) and s (x), then f s ' + s f ' Quotient Rule. If t (x) and b (x), then

26 Summary. Marginal average cost = Marginal average revenue =Marginal average profit =

27 Summary. We learned about higher-order derivatives. That is, derivatives of derivatives. We saw one application of the second derivative.

28 ASSIGNMENT §2.2 on my website 13, 14, 15, 16, 17, 18, 26, 27.