1. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Differentiation 3
Basic Rules of Differentiation
The Product and Quotient Rules
The Chain Rule
Marginal Functions in Economics
Higher-Order Derivatives
Implicit Differentiation and Related Rates
Differentials
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2. Basic Differentiation Rules1.
Ex. 2.
Ex.
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3. Basic Differentiation Rules3.
Ex. 4.
Ex.
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4. More Differentiation Rules5.
Product Rule
Ex.
Derivative of the second function
Derivative of the first function
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5. More Differentiation Rules6.
Quotient Rule
Sometimes remembered as:
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6. More Differentiation Rules6.
Quotient Rule (cont.)
Ex.
Derivative of the denominator
Derivative of the numerator
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7. More Differentiation Rules
7. The Chain Rule
Note: h(x) is a composite function.
Another Version:
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8. More Differentiation Rules
The Chain Rule leads to
The General Power Rule:
Ex.
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Chain Rule Example
Ex.
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10. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Chain Rule Example
Ex.
Sub in for u
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11. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Marginal Functions
The Marginal Cost Function approximates the change in the actual cost of producing an additional unit.
The Marginal Average Cost Function measures the rate of change of the average cost function with respect to the number of units produced.
The Marginal Revenue Function measures the rate of change of the revenue function. It approximates the revenue from the sale of an additional unit.
The Marginal Profit Function measures the rate of change of the profit function. It approximates the profit from the sale of an additional unit.
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Cost Functions
Given a cost function, C(x),
the Marginal Cost Function is
the Average Cost Function is
the Marginal Average Cost Function is
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Revenue Functions
Given a revenue function, R(x),
the Marginal Revenue Function is
Profit Functions
Given a profit function, P(x),
the Marginal Profit Function is
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14. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Elasticity of Demand
If f is a differentiable demand function defined by
Then the elasticity of demand at price p is given by
Demand is: Elastic if E(p) > 1
Unitary if E(p) = 1
Inelastic if E(p) < 1
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Elasticity of Demand
If the demand is elastic at p, then an increase in unit price causes a decrease in revenue. A decrease in unit price causes an increase in revenue.
If the demand is unitary at p, then with an increase in unit price the revenue will stay about the same.
If the demand is inelastic at p, then an increase in unit price causes an increase in revenue. A decrease in unit price causes a decrease in revenue.
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16. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Example
Consider the demand equation
which describes the relationship between the unit price p in dollars and the quantity demanded x of the Acrosonic model F loudspeaker systems. Find the elasticity of demand
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17. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Example
The monthly demand for T-shirts is given by
where p denotes the wholesale unit price in dollars and x denotes the quantity demanded. The monthly cost function for these T-shirts is
1. Find the revenue and profit functions.
2. Find the marginal cost, marginal revenue, and marginal profit functions.
3. Find the marginal average cost function. ……
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18. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Solution
1. Find the revenue and profit functions.
Revenue = xp
Profit = revenue – cost
2. Find the marginal cost, marginal revenue, and marginal profit functions.
Marginal Cost =
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Solution
2. (cont.) Find the marginal revenue and marginal profit functions.
Marginal revenue =
Marginal profit =
3. Find the marginal average cost function.
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20. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Higher Derivatives
The second derivative of a function f is the derivative of the derivative of f at a point x in the domain of the first derivative.
Derivative
Notations
Second
Third
Fourth
nth
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21. Example of Higher Derivatives
Given
find
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22. Example of Higher Derivatives
Given
find
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23. Implicit Differentiation
y is expressed explicitly as a function of x.
y is expressed implicitly as a function of x.
To differentiate the implicit equation, we write f (x) in place of y to get:
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24. Implicit Differentiation (cont.)
Now differentiate
using the chain rule:
which can be written in the form
subbing in y
Solve for y’
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25. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Related Rates
Look at how the rate of change of one quantity is related to the rate of change of another quantity.
Ex.
Two cars leave an intersection at the same time. One car travels north at 35 mi./hr., the other travels west at 60 mi./hr. How fast is the distance between them changing after 2 hours?
Note: The rate of change of the distance between them is related to the rate at which the cars are traveling.
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26. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Related Rates
Steps to solve a related rate problem:
1. Assign a variable to each quantity.
Draw a diagram if appropriate.
2. Write down the known values/rates.
3. Relate variables with an equation.
4. Differentiate the equation implicitly.
5. Plug in values and solve.
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27. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Ex.
Two cars leave an intersection at the same time. One car travels north at 35 mi./hr., the other travels east at 60 mi./hr. How fast is the distance between them changing after 2 hours?
Distance = z y x
From original relationship
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Increments
An increment in x represents a change from x1 to x2 and is defined by:
Read “delta x”
An increment in y represents a change in y and is defined by:
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Differentials
Let y = f (x) be a differentiable function, then the differential of x, denoted dx, is such that
The differential of y, denoted dy, is
Note:
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Example
Given
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31. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Example
The total cost incurred in operating a certain type of truck on a 500-mile trip, traveling at an average speed of v mph, is estimated to be
Find the approximate change in the total operating cost when
the average speed is increased from 55 mph to 58 mph.
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