Hartfield MATH 2040 | Unit 2 Page 1
§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.
Hartfield MATH 2040 | Unit 2 Page 2
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.
Hartfield MATH 2040 | Unit 2 Page 3
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
Hartfield MATH 2040 | Unit 2 Page 4
Example set 2: Find each derivative.
C-1. 3dx
dx
C-2. 4 5f x x
Example set 3: Find each derivative.
D-1. 2
1d
dx x
D-2. 5
1f x
x
Hartfield MATH 2040 | Unit 2 Page 5
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 .)
Hartfield MATH 2040 | Unit 2 Page 6
Example set 1: Find each derivative.
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
Hartfield MATH 2040 | Unit 2 Page 7
Example: Find f x .
623 3 2
6f x x
x
Example: Find f x .
412 6x
f xx
Hartfield MATH 2040 | Unit 2 Page 8
Practice: Find f x .
2
2
88 8f x x
x
Practice: Find f x .
2
5 3f x x
Hartfield MATH 2040 | Unit 2 Page 9
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.
Hartfield MATH 2040 | Unit 2 Page 10
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
Hartfield MATH 2040 | Unit 2 Page 11
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.
Hartfield MATH 2040 | Unit 2 Page 12
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.
Hartfield MATH 2040 | Unit 2 Page 13
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.
Hartfield MATH 2040 | Unit 2 Page 14
§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
Hartfield MATH 2040 | Unit 2 Page 15
Practice: Find f 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
.)
Hartfield MATH 2040 | Unit 2 Page 16
Example: Find f x .
2
( )2 1
xf x
x
Practice: Find f x .
2 1
( )1
xf x
x
Hartfield MATH 2040 | Unit 2 Page 17
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.
Hartfield MATH 2040 | Unit 2 Page 18
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.
Hartfield MATH 2040 | Unit 2 Page 19
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.
Hartfield MATH 2040 | Unit 2 Page 20
§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
Hartfield MATH 2040 | Unit 2 Page 21
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.
Hartfield MATH 2040 | Unit 2 Page 22
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
Hartfield MATH 2040 | Unit 2 Page 23
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
Hartfield MATH 2040 | Unit 2 Page 24
Example: Find .y
2
22 1
xy
x
Practice: Find .y
32 1
4 1
xy
x
Hartfield MATH 2040 | Unit 2 Page 25
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.
Hartfield MATH 2040 | Unit 2 Page 26
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.
Hartfield MATH 2040 | Unit 2 Page 27
§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
Hartfield MATH 2040 | Unit 2 Page 28
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
Hartfield MATH 2040 | Unit 2 Page 29
Example: Find the derivative.
2 2xy x e
Example: Find the derivative.
2
2x
xy
e
Hartfield MATH 2040 | Unit 2 Page 30
Practice: Find the derivative.
x xy xe e
Practice: Find the derivative.
2
1xey
x
Hartfield MATH 2040 | Unit 2 Page 31
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.
Hartfield MATH 2040 | Unit 2 Page 32
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.
Hartfield MATH 2040 | Unit 2 Page 33
§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
Hartfield MATH 2040 | Unit 2 Page 34
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
Hartfield MATH 2040 | Unit 2 Page 35
Example: Find the derivative.
3 lny x x
Example: Find the derivative.
3
2ln
xy
x
Hartfield MATH 2040 | Unit 2 Page 36
Practice: Find the derivative.
2 ln 2xy e x
Practice: Find the derivative.
2
ln 1xy
x
Hartfield MATH 2040 | Unit 2 Page 37
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.
Hartfield MATH 2040 | Unit 2 Page 38
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.