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§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,

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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

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Example 1f (x) = 17

f ‘ (x) = 0

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

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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!

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Examplef (x) = x5

f ‘ (x) = 5 • x4 = 5 x4

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

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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

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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.

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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’.

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Examplef (x) = 7x4

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

f ‘ (x) = 7 • 28 x37 • 4 • x3 =

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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.

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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).

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Example

From the previous examples we get -

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

f ‘ (x) = 15x4 + 4x3 – 6x2 + 10x – 7

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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.

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Notation

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

y ′ f ′ ( x )

)x(fdx

d

dx

yd

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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.

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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

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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.

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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

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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

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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

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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.

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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.

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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.

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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).

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Test Review

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§ 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.

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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

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Test Review

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§ 1.1 Continued

Know integer exponents positive, zero, and negative.

Know fractional exponents.

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Test Review

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§ 1.2

Know the basic business functions

Know functions and the basic terms involved with functions.

Know linear functions.

Know quadratic functions.

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Test Review

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§ 1.3

Know rational functions

Know exponential functions.

Know about shifts to basic graphs.

Know polynomial functions.

Know the difference quotient.

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Test Review

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§ 1.4

Know left and right limits.

Know continuity and the properties of continuity.

Know limits and their properties.

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Test Review

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§ 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

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ASSIGNMENT

§2.1 on my website

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