11 §2.1 Some Differentiation Formulas The student will learn about derivatives of constants, the...

11

§2.1 Some Differentiation Formulas

The student will learn about derivatives

of constants,

the derivative as used in business and economics.

notation, and

of constants, powers, of constants, powers, sums and differences,

22

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

33

Example 1f (x) = 17

f ‘ (x) = 0

If y = f (x) = c then y’ = f ’ (x) = 0.

44

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!

55

Examplef (x) = x5

f ‘ (x) = 5 • x4 = 5 x4

If y = f (x) = xn then y’ = f ’ (x) = n xn – 1.

66

Examplef (x) = 3 x

f (x) = , should be rewritten as f (x) = x1/3 and we can then find the derivative.

3 x

f ‘ (x) = 1/3 x - 2/3

f (x) = x 1/3

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.

88

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’.

99

Examplef (x) = 7x4

If y = f (x) = k • u (x) then f ’ (x) = k • u’.

f ‘ (x) = 7 • 28 x37 • 4 • x3 =

1010

Emphasisf (x) = 7x

If y = f (x) = k • u (x) then f ’ (x) = k • u’.

f ‘ (x) = 7 • 77 • 1 =

REMINDER: If f ( x ) = c x then f ‘ ( x ) = c

The derivative of x is 1.

1111

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).

1212

Example

From the previous examples we get -

f (x) = 3x5 + x4 – 2x3 + 5x2 – 7 x + 4

f ‘ (x) = 15x4 + 4x3 – 6x2 + 10x – 7

1313

Examplef (x) = 3x - 5 - x - 1 + x 5/7 + 5x- 3/5

f ‘ (x) = - 15x - 6 + x - 2 + 5/7 x – 2/7 - 3 x – 8/5

Show how to do fractions on a calculator.

1414

Notation

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

y ′ f ′ ( x )

)x(fdx

d

dx

yd

1515

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.

1616

Example 7

3. Do the above using a graphing calculator.

f (x) = x 2 – 3x

at x = 2.

Using dy/dx under the “calc” menu.

f ’ (x) = 2x – 3

f ’ (2) = 2 2 – 3 = 1

1717

Example 8 - TI-89 ONLY

Do the above using a graphing calculator with a symbolic differentiation routine.

f (x) = 2x – 3x2 and f ’ (x) = 2 – 6x

Using algebraic differentiation under the home “calc” menu.

1818

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

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

1919

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

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

2020

Marginal profit is the derivative of the total profit function and its meaning is the profit generated when producing and selling one more unit.

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

21

Remember – The derivative is -

• The instantaneous rate of change of y with respect to x.

• The limit of the difference quotient.

• The slope of the tangent line.

• h

)x(f)hx(flim

0h

• The 5 step procedure.

• The margin.

2222

Application Example

1. Find the marginal cost at a production level of x radios.

The total cost (in dollars) of producing x portable radios per day is

C (x) = 1000 + 100x – 0.5x2 for 0 ≤ x ≤ 100.

The marginal cost will be C ‘ (x)

C ‘ (x) = 0 + 100 - x

continued

This example shows the essence in how the derivative is used in business.

2323

Example continuedThe total cost (in dollars) of producing x

portable radios per day is

C ‘ (x) = 100 - x

C (x) = 1000 + 100x – 0.5x2 for 0 ≤ x ≤ 100.

2. Find the marginal cost at a production level of 80 radios and interpret the result.

C ‘ (80) =

What does it mean?

100 - 80 = 20

Geometric interpretation!It will cost about $20 to produce the 81st radio.

2424

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).

25

Test Review

25

§ 1.1

Know applied problem involving a straight line

Know the Cartesian plane and graphing.

Know straight lines, slope, and the different forms for straight lines.

2626

ReviewEquations of a Line

General Ax + By = C Not of much use. Test answers.

Slope-Intercept Form y = mx + bGraphing on a calculator.

Point-slope form y – y1 = m (x – x1)

“Name that Line”.

Horizontal line y = b

Vertical line x = a

27

Test Review

27

§ 1.1 Continued

Know integer exponents positive, zero, and negative.

Know fractional exponents.

28

Test Review

28

§ 1.2

Know the basic business functions

Know functions and the basic terms involved with functions.

Know linear functions.

Know quadratic functions.

29

Test Review

29

§ 1.3

Know rational functions

Know exponential functions.

Know about shifts to basic graphs.

Know polynomial functions.

Know the difference quotient.

30

Test Review

30

§ 1.4

Know left and right limits.

Know continuity and the properties of continuity.

Know limits and their properties.

31

Test Review

31

§ 1.5

1. The average rate of change.

h

)x(f)hx(f

2. The instantaneous rate of change.

h

)x(f)hx(flim

0h

3232

ASSIGNMENT

§2.1 on my website

12, 13, 14, 15, 16, 17, 18