§2.1 Some Differentiation Formulas
The student will learn about derivatives
of constants, powers, sums and differences,
notation, and
the derivative as
used in business
and economics.
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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 1
f (x) = 17
If y = f (x) = c then y’ = f ’ (x) = 0.
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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Example
f (x) = x5
If y = f (x) = xn then y’ = f ’ (x) = n xn – 1.
f ‘ (x) = 5 • x4 = 5 x4
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Example
f (x) =
3
x
f (x) = 3 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
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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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Example
f (x) = 7x4
If y = f (x) = k • u (x) then f ’ (x) = k • u’.
f ‘ (x) = 7 • 4 • x3 = 28 x3
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Emphasis
f (x) = 7x
If y = f (x) = k • u (x) then f ’ (x) = k • u’.
f ‘ (x) = 7 • 1 = 7
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
f (x) = 3x5 + x4 – 2x3 + 5x2 – 7 x + 4
From the previous examples we get f ‘ (x) = 15x4 + 4x3 – 6x2 + 10x – 7
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Example
f (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′
d
f ( x)
dx
f′(x)
dy
dx
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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
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
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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.
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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.
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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.
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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.
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Remember – The derivative is • The limit of the difference quotient.
•
lim
h0
f ( x  h)  f ( x)
h
• The instantaneous rate of change of y with
respect to x.
• The slope of the tangent line.
• The 5 step procedure.
• The margin.
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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) = 1000 + 100x – 0.5x2 for 0 ≤ x ≤ 100.
1. Find the marginal cost at a production level of x
radios.
The marginal cost will be C ‘ (x)
C ‘ (x) = 0 + 100 - x
continued
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Example continued
The total cost (in dollars) of producing x
portable radios per day is
C (x) = 1000 + 100x – 0.5x2 for 0 ≤ x ≤ 100.
C ‘ (x) = 100 - 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!
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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
§ 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
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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
Vertical line
y=b
x=a
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Test Review
§ 1.1 Continued
Know integer exponents positive, zero, and negative.
Know fractional exponents.
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Test Review
§ 1.2
Know functions and the basic terms involved with
functions.
Know linear functions.
Know quadratic functions.
Know the basic business functions
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Test Review
§ 1.3
Know polynomial functions.
Know rational functions
Know exponential functions.
Know about shifts to basic graphs.
Know the difference quotient.
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Test Review
§ 1.4
Know limits and their properties.
Know left and right limits.
Know continuity and the properties of continuity.
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Test Review
§ 1.5
1. The average rate of change.
f ( x  h )  f ( x)
h
2. The instantaneous rate of change.
f ( x  h)  f ( x)
lim
h0
h
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ASSIGNMENT
§2.1 on my website
12, 13, 14, 15, 16, 17, 18
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