1. §2.1 Some Differentiation Formulas
The student will learn about derivatives
of constants, powers, sums and differences,
of constants, powers,
of constants,
notation, and
the derivative as used in business and economics. 1

2. The Derivative of a Constant
Let y = f (x) = c be a constant function, then the derivative of the function is
y’ = f ’ (x) = 0.
What is the slope of a constant function?
m = 0 2

3. If y = f (x) = c then y’ = f ’ (x) = 0.
Example 1
f (x) = 17
If y = f (x) = c then y’ = f ’ (x) = 0.
f ‘ (x) = 3

4. THIS IS VERY IMPORTANT. IT WILL BE USED A LOT!
Power Rule.
A function of the form f (x) = xn is called a power function. (Remember √x and all radical functions are power functions.)
Let y = f (x) = xn be a power function, then the derivative of the function is
y’ = f ’ (x) = n xn – 1.
THIS IS VERY IMPORTANT. IT WILL BE USED A LOT! 4

5. If y = f (x) = xn then y’ = f ’ (x) = n xn – 1.
Example
f (x) = x5
If y = f (x) = xn then y’ = f ’ (x) = n xn – 1.
f ‘ (x) =
5 • x4 =
5 x4 5

6. Example
f (x) =
f (x) = , should be rewritten as f (x) = x1/3 and we can then find the derivative.
f (x) = x 1/3
f ‘ (x) =
1/3 x - 2/3 6

7. Derivative of f (x) = x
The derivative of x is used so frequently that it should be remembered separately.
This result is obvious geometrically, as shown in the diagram.

8. Constant Multiple Property.
Let y = f (x) = k • u (x) be a constant k times a differential function u (x). Then the derivative of y is
y’ = f ’ (x) = k • u’ (x) = k • u’. 8

9. If y = f (x) = k • u (x) then f ’ (x) = k • u’.
Example
f (x) = 7x4
If y = f (x) = k • u (x) then f ’ (x) = k • u’.
f ‘ (x) =
7 • 4 • x3 =
7 •
28 x3 9

10. If y = f (x) = k • u (x) then f ’ (x) = k • u’.
Emphasis
f (x) = 7x
If y = f (x) = k • u (x) then f ’ (x) = k • u’.
f ‘ (x) =
7 •
7 • 1 = 7
REMINDER: If f ( x ) = c x then f ‘ ( x ) = c
The derivative of x is 1. 10

11. Sum and Difference Properties.
The derivative of the sum of two differentiable functions is the sum of the derivatives.
The derivative of the difference of two differentiable functions is the difference of the derivatives. OR
If y = f (x) = u (x) ± v (x), then
y ’ = f ’ (x) = u ’ (x) ± v ’ (x). 11

12. From the previous examples we get -
f (x) = 3x5 + x4 – 2x3 + 5x2 – 7 x + 4
From the previous examples we get -
f ‘ (x) =
15x4
+ 4x3
– 6x2
+ 10x
– 7 12

13. Example f (x) = 3x - 5 - x - 1 + x 5/7 + 5x- 3/5 f ‘ (x) = - 15x - 6
Show how to do fractions on a calculator. 13

14. Notation
Given a function y = f ( x ), the following are all notations for the derivative.
y ′
f ′ ( x ) 14

15. Graphing Calculators
Most graphing calculators have a built-in numerical differentiation routine that will approximate numerically the values of f ’ (x) for any given value of x.
Some graphing calculators have a built-in symbolic differentiation routine that will find an algebraic formula for the derivative, and then evaluate this formula at indicated values of x. 15

16. Example 7
f (x) = x 2 – 3x
at x = 2.
3. Do the above using a graphing calculator.
Using dy/dx under the “calc” menu.
f ’ (x) = 2x – 3
f ’ (2) = 2  2 – 3 = 1 16

17. Example 8 - TI-89 ONLY f (x) = 2x – 3x2 and f ’ (x) = 2 – 6x
Do the above using a graphing calculator with a symbolic differentiation routine.
Using algebraic differentiation under the home “calc” menu. 17

18. 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)
Marginal cost is the derivative of the total cost function and its meaning is the additional cost of producing one more unit. 18

19. 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)
Marginal revenue is the derivative of the total revenue function and its meaning is the revenue generated when selling one more unit. 19

20. Marginal Profit
If x is the number of units of a product produced and sold in some time interval, then
Total profit = P = R (x) – C (x)
Marginal profit = P ’ (x) = R’ (x) – C’ (x)
Marginal profit is the derivative of the total profit function and its meaning is the profit generated when producing and selling one more unit. 20

21. Remember – The derivative is -
The limit of the difference quotient.
The instantaneous rate of change of y with respect to x.
The slope of the tangent line.
The 5 step procedure.
The margin.

22. The total cost (in dollars) of producing x portable radios per day is
Application Example
This example shows the essence in how the derivative is used in business.
The total cost (in dollars) of producing x portable radios per day is
C (x) = x – 0.5x2 for 0 ≤ x ≤ 100.
Find the marginal cost at a production level of x radios.
The marginal cost will be C ‘ (x)
- x
C ‘ (x) =
+ 100
continued 22

23. The total cost (in dollars) of producing x portable radios per day is
Example continued
The total cost (in dollars) of producing x portable radios per day is
C (x) = x – 0.5x2 for 0 ≤ x ≤ 100.
C ‘ (x) = x
2. Find the marginal cost at a production level of 80 radios and interpret the result.
C ‘ (80) =
100
- 80 = 20
What does it mean?
It will cost about $20 to produce the 81st radio.
Geometric interpretation! 23

24. Summary. If f (x) = C then f ’ (x) = 0.
If f (x) = xn then f ’ (x) = n xn – 1.
If f (x) = k • u (x) then f ’ (x) = k • u’ (x) = k • u’.
If f (x) = u (x) ± v (x), then
f ’ (x) = u’ (x) ± v’ (x). 24

25. Test Review § 1.1 Know the Cartesian plane and graphing.
Know straight lines, slope, and the different forms for straight lines.
Know applied problem involving a straight line 25

26. Review Equations of a Line
General Ax + By = C Not of much use. Test answers.
Slope-Intercept Form y = mx + b Graphing on a calculator.
Point-slope form y – y1 = m (x – x1) “Name that Line”.
Horizontal line y = b
Vertical line x = a 26

27. Test Review § 1.1 Continued
Know integer exponents positive, zero, and negative.
Know fractional exponents. 27

28. Test Review
§ 1.2
Know functions and the basic terms involved with functions.
Know linear functions.
Know quadratic functions.
Know the basic business functions 28

29. Test Review § 1.3 Know polynomial functions. Know rational functions
Know exponential functions.
Know about shifts to basic graphs.
Know the difference quotient. 29

30. Test Review § 1.4 Know limits and their properties.
Know left and right limits.
Know continuity and the properties of continuity. 30

31. Test Review § 1.5 The average rate of change.
2. The instantaneous rate of change. 31

32. ASSIGNMENT §2.1 on my website 12, 13, 14, 15, 16, 17, 1832