1. Chain Rule: Power Form Marginal Analysis in Business and Economics
The student will learn about:
the chain rule, combining different rules of derivation, and an application.
Marginal cost, revenue, and profit as well as,
applications, andmarginal average cost,
revenue and profit.
The slides for this text are organized into chapters. This lecture covers Chapter 1.
Chapter 1: Introduction to Database Systems
Chapter 2: The Entity-Relationship Model
Chapter 3: The Relational Model
Chapter 4 (Part A): Relational Algebra
Chapter 4 (Part B): Relational Calculus
Chapter 5: SQL: Queries, Programming, Triggers
Chapter 6: Query-by-Example (QBE)
Chapter 7: Storing Data: Disks and Files
Chapter 8: File Organizations and Indexing
Chapter 9: Tree-Structured Indexing
Chapter 10: Hash-Based Indexing
Chapter 11: External Sorting
Chapter 12 (Part A): Evaluation of Relational Operators
Chapter 12 (Part B): Evaluation of Relational Operators: Other Techniques
Chapter 13: Introduction to Query Optimization
Chapter 14: A Typical Relational Optimizer
Chapter 15: Schema Refinement and Normal Forms
Chapter 16 (Part A): Physical Database Design
Chapter 16 (Part B): Database Tuning
Chapter 17: Security
Chapter 18: Transaction Management Overview
Chapter 19: Concurrency Control
Chapter 20: Crash Recovery
Chapter 21: Parallel and Distributed Databases
Chapter 22: Internet Databases
Chapter 23: Decision Support
Chapter 24: Data Mining
Chapter 25: Object-Database Systems
Chapter 26: Spatial Data Management
Chapter 27: Deductive Databases
Chapter 28: Additional Topics
Dr .Hayk Melikyan
Departmen of Mathematics and CS

2. Chain Rule: Power Rule.
We have already made extensive use of the power rule with xn,
We wish to generalize this rule to cover [u (x)]n.
That is, we already know how to find the derivative of
f (x) = x 5
We now want to find the derivative of
f (x) = (3x 2 + 2x + 1) 5

3. * * * * * VERY IMPORTANT * * * * *
Chain Rule: Power Rule.
General Power Rule. [Chain Rule]
Theorem 1. If u (x) is a differential function, n is any real number, and
y = f (x) = [u (x)]n
then
f ’ (x) = n[ u (x)]n – 1 u’ (x) = n un – 1u’ or
* * * * * VERY IMPORTANT * * * * *

4. Find the derivative of y = (x3 + 2) 5.
Example 1
Find the derivative of y = (x3 + 2) 5.
Chain Rule
NOTE: If we let u = x 3 + 2, then y = u 5.
Let u (x) = x3 + 2, then y = u 5 and u ‘ (x) =
3x2 5
(x3 + 2) 4
3x2
= 15x2(x3 + 2)4

5. Find the derivative of y =
Example 2
Find the derivative of y =
Rewrite as y = (x 3 + 3) 1/2
Then y’ = 1/2 (x 3 + 3) – 1/2
Then y’ = 1/2
Then y’ = 1/2 (x 3 + 3) – 1/2 (3x2)
= 3/2 x2 (x3 + 3) –1/2

6. Combining Rules of Differentiation
The chain rule just developed may be used in combination with the previous rules for taking derivatives
Some examples follow.

7. Example 3 Find f ’ (x) if f (x) =
We will use a combination of the quotient rule and the chain rule.
Let the top be t (x) = x4, then t ‘ (x) =
4x3
Let the bottom be b (x) = (3x – 8)2, then using the chain rule b ‘ (x) =
2 (3x – 8) 3 =
6 (3x – 8)

8. Example 4
Find f ’ (x) and find the equation of the line tangent to the graph of f at the indicated value of x.
f (x) = x2 (1 – x)4; at x = 2.
We will use the point-slope form. The point will come from (2, f(2)) and the slope from f ‘ (2).
Point -
When x = 2, f (x) = 22 (1 – 2)4 = (4) (1) = 4
Hence the tangent goes through the point (2,4).
f ‘ (x) = x2
4 (1 – x)3 (-1) +
(1 – x)4 2x
= - 4x2 (1 – x)3 + 2x(1 – x)4 and
f ‘ (2) = (- 4) (4) (-1)3 + (2) (2) (-1)4 = = 20 = slope

9. Example 4 continued
Find f ’ (x) and find the equation of the line tangent to the graph of f at the indicated value of x.
f (x) = x2 (1 – x)4; at x = 2.
We will use the point-slope form. The point is (2, 4) and the slope is 20.
y – 4 = 20 (x – 2) = 20x – 40 or
y = 20x

10. By graphing calculator!
Example 4 continued
Find f ’ (x) and find the equation of the line tangent to the graph of f at the indicated value of x.
f (x) = x2 (1 – x)4; at x = 2.
By graphing calculator!
Graph the function and use “Math”, “tangent”.
-2 ≤ x ≤ 3
-1 ≤ y ≤ 20

11. Application
P. 202, #78. The number x of stereo speakers people are willing to buy per week at a price of $p is given by
x = 1,
for 20 ≤ p ≤ 100
Find dx/dp.
f ‘ (p) =
- (60)
(1/2)
(p + 25)-1/2
(1)

12. Application continued
The number x of stereo speakers people are willing to buy per week at a price of $p is given by
x = 1,
for 20 ≤ p ≤ 100
2. Find the demand and the instantaneous rate of change of demand with respect to price when the price is $75.
That is, find f (75) and f ‘ (75).
f (75) = 1,000 – 60
= 1000 – 600 = 400
f ‘ (75)
= -30/10 = - 3

13. Application continued
The number x of stereo speakers people are willing to buy per week at a price of $p is given by
x = 1,
for 20 ≤ p ≤ 100
3. Give a verbal interpretation of these results.
With f (75) = 400 and f ‘ (75) = - 3 that means
that the demand at a price of $75 is 400 speakers and
each time the price is raised $1, three fewer speakers are purchased.

14. Summary. If
y = f (x) = [u (x)]n
then

15. Marginal Cost, Revenue, and Profit
Remember that margin refers to an instantaneous rate of change, that is, a derivative.
Marginal Cost
If x is the number of units of a product produced in some time interval, then
Total cost = C (x)
Marginal cost = C’ (x)

16. Marginal Cost, Revenue, and Profit
Marginal Revenue
If x is the number of units of a product sold in some time interval, then
Total revenue = R (x)
Marginal revenue = R’ (x)

17. Marginal Cost, Revenue, and Profit
Marginal Profit
If x is the number of units of a product produced and sold in some time interval, then
Total profit = P (x) = R (x) – C (x)
Marginal profit = P’ (x) = R’ (x) – C’ (x)

18. Marginal Cost and Exact Cost.
Theorem C (x) is the total cost of producing x items and
C (x + 1) is the cost of producing x + 1 items.
Then the exact cost of producing the x + 1st item is
C (x + 1) – C (x)
The marginal cost is an approximation of the exact cost. Hence,
C ’ (x) ≈ C (x + 1) – C (x).
The same is true for revenue and profit.

19. Example 1
P. 210, #2. The total cost of producing x electric guitars is
C (x) = 1, x – 0.25 x2
Find the exact cost of producing the 51st guitar.
Exact cost is C (x + 1) – C (x)
C (51) = $
C (50) = $
Exact cost = $ $5375 = $74.75

20. Example 1 continued The total cost of producing x electric guitars is
C (x) = 1, x – 0.25 x2
2. Use the marginal cost to approximate the cost of producing the 51st guitar.
The marginal cost is C ‘ (x)
C ‘ (x) =
100 – 0.5x
C ‘ (50) =
$75.00
Exact cost = $ $5375 = $74.75

21. 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 =

22. 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 =

23. 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 =

24. 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

25. Example 2
P. 210, # 4. 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

26. Example 2 continued Marginal average cost =
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?

27. Example 2 continued Average cost = $30.00
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

28. Example 3
P. 211, #14. The price-demand equation and the cost function for the production of television sets are given, respectively by
p (x) = 300 -
and
C (x) = 150, x
where x is the number of sets that can be sold at a price of $p per set and C (x) is the total cost of producing x sets.
Find the marginal cost.
The marginal cost is C ‘ (x) so
What does this mean?
C ‘ (x) = $30.

29. Example 3 continued
The price-demand equation and the cost function for the production of television sets are given, respectively by
p (x) = 300 -
and
C (x) = 150, x
2. Find the revenue function in terms of x.
The revenue function is R (x) = x · p (x), so

30. Example 3 continued
The price-demand equation and the cost function for the production of television sets are given, respectively by
and
C (x) = 150, x
3. Find the marginal revenue.
The marginal revenue is R ‘ (x), so

31. Example 3 continued
The price-demand equation and the cost function for the production of television sets are given, respectively by
and
C (x) = 150, x
4. Find R’ (1,500) and interpret the results.
What does this mean?
At a production rate of 1,500 sets, revenue is increasing at the rate of about $200 per set.

32. Example 3 continued
The price-demand equation and the cost function for the production of television sets are given, respectively by
C (x) = 150, x
and
5. Graph the cost function and the revenue function on the same coordinate. Find the break-even point.
0 ≤ x ≤ 9,000
0 ≤ y ≤ 700,000
(600,168000)
(7500, )

33. Example 3 continued
The price-demand equation and the cost function for the production of television sets are given, respectively by
C (x) = 150, x
and
6. Find the profit function in terms of x.
The profit is revenue minus cost, so

34. Example 3 continued
The price-demand equation and the cost function for the production of television sets are given, respectively by
7. Find the marginal profit.
The marginal profit is P ‘ (x), so

35. Example 3 continued
The price-demand equation and the cost function for the production of television sets are given, respectively by
7. Find P’ (1,500) and interpret the results.
What does this mean?
At a production level of 1500 sets, profit is increasing at a rate of about $170 per set.

36. Summary.
In business the instantaneous rate of change, the derivative, is referred to as the margin.