§4.1 Basic Techniques for Finding Derivatives There are...

Hartfield MATH 2040 | Unit 2

§4.1 Basic Techniques for Finding Derivatives In the previous unit we introduced the mathematical concept of the derivative:

0

( ) ( )limh

f x h f xf x

h (assuming the limit exists)

In this unit we will look at rules for finding derivatives that will be simpler than applying the definition.

There are many different forms of notation used to indicate a derivative. For example, if we want the derivative of the function f(x), we could express it as: f x (prime notation)

d

f xdx

(Leibniz notation)

xD f x (subscript notation)

Assuming that y = f(x), a common variation on

Leibniz notation is dy

dx.


Example: Express all appropriate forms of

notation that could be used for the derivative of the function g(t).

The notation d

dx may sometimes be used to

indicate that you wish to find a derivative without defining an expression explicitly as a function.


Rules For Differentiation Rule 1: Constant Rule

For any real constant k,

0d

kdx

.

(If f x = k, then f x = 0.)

Rule 2: Power Rule

For any constant exponent n,

1n ndx n x

dx .

(If f x = nx , then f x = 1nn x .)

Example set 1: Find each derivative.

A-1. 42d

dx

A-2. 13f x

B-1. 4dx

dx

B-2. 7f x x



C-1. 3dx

dx

C-2. 4 5f x x


D-1. 2

1d

dx x

D-2. 5

1f x

x


Practice: Find the derivative of each function.

A. 12f x x

B. f x x

C. 3

1f x

x

Rule 3: Constant Multiple Rule

For any constant k,

d

k f x k f xdx

.

(If f x = k g x ,

then f x = k g x .)

Rule 4: Sum/Difference Rules

d

f x g x f x g xdx

(If f x = u x v x ,

then f x = u x v x .)



A-1. 510d

xdx

A-2. 86f x x

B-1. 3 2dx x

dx

B-2. 6 5f x x x

Two noteworthy shortcuts based on rules 2 and 3:

1d

xdx

and d

c x cdx

Example set 2: Find the derivative.

C-1. 7d

xdx

C-2. 4 85f x x x


Example: Find f x .

623 3 2

6f x x

x

Example: Find f x .

412 6x

f xx


Practice: Find f x .

2

2

88 8f x x

x


2

5 3f x x


Marginal Analysis We will use the following function notations for application problems in business and economics: Revenue Function R(x) = Total revenue from selling x units Cost Function C(x) = Total cost of producing x units Profit Function P(x) = Total profit from producing and selling x units

Economists use the term marginal to refer to rates of change. The derivative, which coincides with instantaneous rate of change, is used when talking about marginal in calculus. If you have a cost function at some level of production x, the marginal cost is the expected additional cost of producing the (x + 1)st unit. That is, if you are already making x units, the marginal cost predicts the cost of making the next unit. Notationally, when C(x) represents the cost function, C′(x) represents the marginal cost function.


It is important to understand that the marginal function may not exactly identify the additional cost of the (x + 1)st unit. The actual cost of that next unit can be exactly found by calculating C(x + 1) – C(x). In most cases though, the evaluation of the marginal cost function at x is very close to the exact value found by the difference. Analogous statements can be made for revenue and profit.

Some additional notes for future reference:

Analogous statements can be made for revenue and profit. Thus R′(x) represents the marginal revenue function and P′(x) represents the marginal profit function.

A common function in economics is the demand function p D q which relates the

number of units q that consumers are willing to purchase at price p. The revenue generated from selling q units is then found by .R q q D q


Example: A steel mill determines that its cost

function is 3( ) 8000 6000C x x x dollars,where x is in the daily production of tons of steel.

A. Find the cost of manufacturing 64 tons

of steel per day.

B. Find the marginal cost function.

C. Find the marginal cost of producing one more ton when 64 tons are being produced.

D. Calculate the actually cost of producing one more ton by finding the cost of manufacturing 65 tons.


Example: If the demand function for

heavyweight paper is 500

25

qp

dollars, where q is in reams, answer the following:

A. Find the revenue function.

B. Find the revenue generated from 200 reams being sold.

C. Find the marginal revenue function.

D. Calculate and interpret the marginal revenue function when 200 reams are being sold.


Practice: Continuing the previous example,

suppose the cost function in dollars for heavyweight paper is given by

( ) 210 4 , 0 300.C q q q

A. Find the profit function.

B. Find the marginal profit function.

C. Calculate and interpret the marginal profit function when 100 reams, 200 reams, and 250 reals are being produced and sold.


§4.2 Derivatives of Products and Quotients Rule 5: Product Rule

df x g x

dx

f x g x g x f x

(If f x = u x v x ,

then f x = u x v x v x u x .)

Example: Find f x .

2( ) 4f x x x x



2( ) 2 5 3f x x x

Rule 6: Quotient Rule

2

f x g x f x f x g xd

dx g x g x

(If f x =

u x

v x and 0v x ,

then f x =

v x u x u x v x

v x

.)


Example: Find f x .

2

( )2 1

xf x

x


2 1

( )1

xf x

x


Average Cost and Marginal Average Cost For a cost function C where C x represents

the cost of manufacturing x items, the average

cost function C x is found by /C x x and

determines the average cost per item. It is possible to find a marginal with respect to an average function. As with other marginals, a marginal average function is predictive of the change occurring when you increase the number of items by one.

The marginal average cost function C x is

the derivative of the average cost function and finds the rate of change in the average cost. Analogous statements can be made for

revenue and profit. Thus R x represents the

average revenue function, determined by

/ ,R x x with R x representing the marginal

average revenue function. P x represents

the average profit function, determined by

/ ,P x x with P x representing the marginal

average revenue function.


Example: The total profit (in tens of dollars)

from selling x self-help books is 5 6

( ) .2 3

xP x

x

A. Find the average profit function.

B. Find the marginal average profit function.

C. Evaluate P , P , and P when x = 20. Interpret each evaluation.


Practice: The fuel economy (in m.p.g.) of a

Porsche driven at a speed of x m.p.h.

is 2

2000( ) .

3025

xE x

x

A. Find ( )E x

B. Evaluate E and E when x = 80, rounding logically. Interpret each evaluation.


§4.3 The Chain Rule Recall from algebra that composed functions consist of one function inside a second

function. For example, 32 1x is considered

to be a composite function because the function x² – 1 exists within a cubing function (that is, the output of x² – 1 is the input to the

cube). We can decompose 32 1x by

defining the two functions that are brought together to make the new function.

32 1 ( )x f g x with

3

2

( )

( ) 1

f x x

g x x

We tend to call g the “inside function” and f the “outside function”. Example: Decompose each of the following

functions so that f(g(x)) returns the original function.

63x x


Practice: Decompose each of the following

functions so that f(g(x)) returns the original function.

A. 4

2 1x

B. 2 7x

Finding a derivative for a function created by a composition requires we consider the differentiation of both the inside and outside function. It turns out that the outside derivative is taken initially without regard to the inside expression, with the inside derivative being multiplied in separately. This is called the Chain rule.


Chain Rule Rule 7: Chain Rule

( ) ( ) ( )d

f g x f g x g xdx

We can also restate the composition

( )y f g x as y = f(u) and u = g(x).

Their respective derivatives would be

( )dy

f udu

and ( )du

g xdx

.

By appropriate substitution, we can present Chain Rule using Leibniz’s Notation:

.dy dy du

dx du dx

Example: Find .y

A. 63y x x

B. 2 7y x


Practice: Find .y

A. 4

2 1y x

B.

32

1

2y

x x

Frequently you may need to combine Chain Rule with Product Rule or Quotient Rule to find a derivative. Example: Find .y

42 3 1y x x


Example: Find .y

2

22 1

xy

x

Practice: Find .y

32 1

4 1

xy

x


Applications Example: Suppose that for a group of 10,000

people, the number who survive to

age x is ( ) 1000 100N x x . Evaluate and interpret N and N′ when x = 36.


Practice: A 35 year old male of average weight

is injected with a 100 cubic centimeters of a specific medication. At t hours after injection, the body is

metabolizing

3

200( )

1

tV t

t

cc of the

medication. Evaluate and interpret V and V′ at t = 4.


§4.4 Derivatives of Exponential Functions Rule 8: Exponential Rule (if base is e)

x xde e

dx

Rule 8*: Exponential Rule (if base is a)

lnx xda a a

dx

Our primary but not exclusive focus will be on differentiating exponential expressions with a base of e.

Frequently you will need to find derivatives where the exponent of a base e (or a) exponential is not simply x. Strictly speaking this creates a composition and requires the use of Chain Rule. We can integrate Chain Rule with the Exponential Rule as follows: Rule 8a:

g x g xde e g x

dx

lng x g xd

a a a g xdx


Example: Find the derivative of each.

A. 4 110 xy e

B. 342

3xy e

C. 35 2 xy

Practice: Find the derivative of each.

A. 65 xy e

B. 26 38 xy e

C. 124 3

xy


Example: Find the derivative.

2 2xy x e


2

2x

xy

e


Practice: Find the derivative.

x xy xe e


2

1xey

x


Applications: Example: A cup of coffee brewed at 200

degrees, if left in a 70-degree room, will cool to T(t) = 70 + 130e –0.04t (°F) in t minutes.

Determine the temperature of the

coffee in 1 hour and the rate of change in the temperature at that time.


Example: For a particular market the demand

function of an item is 0.1200 ,qp e where q is in thousands of units.

Find the revenue function and its

derivative. Then evaluate both and interpret when 5 thousand units are being sold.


§4.5 Derivatives of Logarithmic Functions Rule 9: Logarithmic Rule (if base is e, x > 0)

1

lnd

xdx x

Rule 9*: Logarithmic Rule (if base is a, x > 0)

1log

lna

dx

dx a x

Similar to exponentials, our primary but not exclusive focus will be on differentiating logarithmic expressions with a base of e.

Recall that the domain of a logarithmic function is based on when the argument of the log is positive. As with exponentials, frequently you will need to find derivatives where the argument is not simply x. Again this creates a composition and requires the use of Chain Rule. We can integrate Chain Rule with the Logarithmic Rule as follows: Rule 9a:

ln ,

g xdg x

dx g x

where g > 0

log

ln( )a

g xdg x

dx a g x


Example: Find the derivative of each.

A. ln 8 3y x

B. 3

22ln 5y x

C. log 4y x

Practice: Find the derivative of each.

A. 3ln 1y x

B. 22log 2y x x



3 lny x x


3

2ln

xy

x



2 ln 2xy e x


2

ln 1xy

x


Applications: Example: The total revenue (in thousands of

dollars) produced by selling x thousands of books can be expressed as ( ) 50ln 4 1R x x .

The cost (in thousands of dollars) to produce x thousands of book is given by ( ) 5 .C x x

A. Find the marginal revenue function and

interpret it when 10 thousand books are being sold.

B. Find the profit function and the marginal profit function. Interpret both when 10 thousand books are being sold.


Example: Based on projections from the Kelly

Blue Book, the average resale value of a 2010 Toyota Corolla sedan can be anticipated by the function

( ) 15450 13915log 1 ,f t t where

t is the number of years since 2010. Find & interpret f and f′ when t = 4.

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