Transcript Derivatives and Differentiation

Derivatives and Differentiation
Chapter 6
6.1 Introduction
Slopes of the function graphs are important in economics
1) Total cost curves: how fast do costs increase with production?
2) Demand curves: how sensitive are consumers to changes in price?
3) PPC curves: how much bread must be given up in order to produce more milk?
Definition. The technique of measuring the slope of a function (in case it exists) is
called differentiation. It is also known as differential calculus, or just calculus.
Note: differentiation is about more than just measuring slopes, but at this stage the ‘slope’ definition will
just suffice.
Objective: learn how to calculate slopes.
6.2 Difference Quotient
We use Greek letter
delta to denote changes in variables.
Definition. Consider two variable values y0 and y1 . The difference y  y1  y0
is called an increment of variable y.
General Setting
Since equilibrium values of the endogenous variables can be represented as
functions of the exogenous variables, it would be handy to give an answer to
the following general question: given a change in exogenous variable x,
what would be the resulting change in the endogenous variable y?
y  f a 
a
Current value of the
exogenous variable x
y Corresponding value of
the endogenous variable y
a  x  y  y  f a    f a  x   f a 
A disturbance to the
exogenous variable
The resulting change in the
endogenous variable
New value of the
endogenous
variable
Difference Quotient
A change in the initial value of x  aby x results in a corresponding change in
the level of y by y  f a  x   f a 
Definition. Consider a function y=f(x). A change in y per unit of a change in x is
called difference quotient, and is computed as follows:
y
f a  x   f a 

x
x
Note 1. Difference quotient is also known as Newton quotient.
Note 2. Difference quotient represents the amount of change in y per unit of change in
x, so that it can be interpreted as the rate of change of y with respect to changes in x.
Note 3. In case the increase in x results in a decrease in y the difference quotient is
negative.
6.3 Calculating the Difference Quotient
Consider the following quadratic function:
y  f x  3x2  4
Initial value of x:
a
Initial value of y:
y0  f a  3a  4
The new value of y:
2
y0  y  f a  x   3a  x   4
2
y 3a  x   4  3a   4 6ax  3x 


x
x
x
2
The difference quotient is:
2
y 6a  3x

 6a  3x
x
1
2
6.4 The Slope of a Curved Line
It is often important to know exactly how steep a curve is.
For the linear function, we know that its slope is equal to a single parameter p if the
function is
y  px  q
The precise value of the slope of any function is given by what we will call a derivative
function: f  a = the slope of the tangent to the curve f(x)at point (a,f(a)).
 
f a1 
f a1  is the slope of the tangent
line to the graph of function
f(x) at point a
1
a1
Difference Quotient and the Slope
Definition. Consider any two points P and Q on a curve. A line segment connecting
points P and Q is called a chord.
The difference quotient is measuring the slope of a chord connecting P and Q.
Fixing point P, there exist infinitely many points Q, and hence infinitely many
chords and the associated slopes. Which one is the slope of the curve at point P?
Tangents and Slopes
Consider the following cost function: CQ  1  Q2
C
A straight line passing
through A and B
C1
is crossing the horizontal
axis
at angle 
B
C
Q  0 Q
tan   lim
C
whose tangent
C
is equal to
tan  
Q0
 C0 C

Q1  Q0 Q
1
A
C0

As Q is getting smaller, this straight line will be
approaching the tangent line to C(Q) at
Q0

Q0
Q
Q1
Q
Tangent Line
Definition. A straight line that has only one
single point in common with a curve is
called a tangent to the curve at that point.
Straight line AB is tangent to the curve
y  x 2  4 at point P   x0 , y0  .
6.5, 6.6 The Derivative
Definition. Consider a function f  x  at point x0 . Consider an increment of the
function’s argument x taken at point x0. The limit of a difference quotient
computed for function f x  at point x0 for x  0 is called a derivative of
function f at point x0 .
Notation:
dy
dx
y
x  0 x
 f a   lim
a
 lim
a
x  0
f a  x   f a 
x
 
Note 1. The derivative f  x taken at point x0 is a function of point x0 alone,
while the difference quotient is a function of both x0 and x .
Note 2. Derivatives may not be computed for certain functions. For
example, functions with kinks do not have derivatives everywhere in
their domain.
Note 3. When computing a derivative, remember it only makes
sense to do so with a particular point in the function’s domain in
mind—”loose” derivatives are nonsensical.
A Straightforward Recipe for
Computing the Derivative
A) Choose point a in the function’s domain and increase it by a (small) x
B) Compute the corresponding change in the function’s value: f a  x   f a 
C) Compute the difference quotient f a  x  f a



x
D) Try to simplify the Newton quotient as much as possible
E) Infer the value of the derivative as the value of the Newton Quotient for very small x
Straightforward Recipe: an Example
Denote h  x .
A)
f a  h  a  h  a3  3a2h  3ah2  h3
B)
f a  h  f a  a3  3a 2h  3ah2  h3  a3  3a2h  3ah2  h3
3

C-D)

f a  h   f a  3a 2 h  3ah2  h3

 3a 2  3ah  h 2
h
h
E) As h grows smaller, the difference quotient above “loses” 3ah and
so that its value becomes equal to
h2
3a 2
While the straightforward recipe works all right on simple functions, it becomes very
difficult to apply it on more complicated functions like
y  3x 2  x  1
6.7 Constant-Function Rule
Consider
y  f x   k
def
dy
f a  x   f a 
k k
 f a   lim
 lim
0
x 0
x 0 x
dx a
x
!
There is an important difference between f x   0 and f a   0
f x   0 means that the derivative of function f is equal to zero in the domain of
f—in that case function f is a constant.
f a   0 means that function f has a zero derivative at a particular point a
--in that case the tangent to function f’s graph is parallel to the horizontal
axis at that point.
Power Rule
In case a is an arbitrary constant,
f x  x  f x  ax
a 1
a
For example, consider
f x   x 3


f a  h   f a   a 3  3a 2 h  3ah2  h 3  a 3 
 3a 2 h  3ah2  h 3
f a  h   f a 
 3a 2  3ah  h 2  3a 2
h0
h
Clearly, differentiating a polynomial of degree n the number of times equal to n+1
will result in a function that is identically equal to 0.
Additive and Multiplicative Constants
Additive constant rule:
y  A  f x  y  f x
Remember adding a constant means shifting the graph vertically up or
down, which doesn’t change the slope.
Multiplicative constant rule:
y  Af x   y  Af x 
This rule follows from the constant function rule and the definition of a
derivative.
Sums, Products, and Quotients
Suppose we know how to differentiate each one of the two functions f x 
and g x  . How do we differentiate, say, f x  g x ?
Sum-difference rule:
d
 f x   g x   f x   g x 
dx
Note that from now on we will not indicate at a specific point x0 at which a
derivative is taken, unless we really need to.
That is, we are going to write f  x  assuming f x0 
Product rule:
Quotient rule:
d
 f x g x   f x g x   f x g x 
dx
d  f x   f x g x   f x g x 


dx  g x  
g 2 x 
Sum-Difference Rule
d
 f x   g x   f x   g x 
dx
This rule can be extended to any number of summand functions:
d
 f x   g x   hx   ...  f x   g x   hx   ...
dx
For example,


d
7 x 4  2 x 3  3 x  37  28 x 3  6 x 2  3
dx
The constant in this equation drops out
when we differentiate the function.
Product Rule
d
dx
 f  x g  x  
By definition of
the derivative,
f  x g  x   f  x g  x 
def
d
f  x  x g  x  x   f  x g  x 
 f x g x   lim
x 0
dx
x
Add and subtract
f x g x  x 
in the numerator:
 lim
Then collect the
terms:
 lim
x 0
x 0

f  x  x g  x  x   f  x g x  x   f  x g  x   f  x g  x  x 
x
 f x  x   f x g x  x   lim f x g x  x   g x 
x
x  0
 f x g x   f x g x 
x

Marginal and Average Revenues
Suppose we know that a firm’s average revenue is given by some function
ARQ  f Q What can we say about the difference between average and
marginal revenues?
We can apply the product rule of differentiation to answer that question.
def
MRQ  
d
d
ARQ Q  d  f Q Q  f QQ  f Q
TRQ  
dQ
dQ
dQ
Marginal revenue is by definition a derivative of the total revenue function
which in turn is equal to the product of average revenue and quantity.
In case of perfect competition we know that average revenue is the same (since the
competitive price is constant), so f Q  0 and the marginal revenue is equal to the
average revenue.
On the other hand, when the average revenue (i.e. the demand curve) is downward
sloping, MRQ  ARQ  f QQ  0 reflecting the fact that under imperfect
competition, the marginal revenue curve always lies below the demand curve.
MRQ  ARQ  f QQ  f Q  f Q  f QQ
Primitive and Marginal Functions
Consider a function f  x .
Definition. A derivative function
f x  is called a marginal function of function f(x).
Definition. Function f x  is called a primitive function with respect to the marginal
function f  x .

Example. Consider a demand function P  f Q  where P is the price, and Q is
quantity. Revenue is defined as the product R Q  P  Q  f Q Q .
 
 
The revenue function R(Q) is a primitive function with respect to the marginal
revenue function defined as d RQ  .
dQ
Quotient Rule
d  f x  
f x g x   f x g x 




dx 
g 2 x 
 g x  
By definition:
Reducing to a
common
denominator:
Adding and
subtracting
f xg x
Rearranging:
f x  x 
d  f x  

  lim
dx  g x   x0
 lim
x 0
g x  x 
x
 f x 
f x  x g x   f x g x  x  1
g x  x g x 
x
g x 


1 f x  x g x   f x g x   f x g x  x   f x g x 
x 0 x
g x  x g x 
 lim
1
f x  x   f x 
g x  x   g x 





g
x

f
x

x 0 g  x g  x  x  
x
x

 lim
1
g x 
2
f  x 
g  x 

An Example
Consider a derivative of the following rational function:
a x2  b
z x  
cx
Identify functions f  x  and g  x  :
f  x   ax2  b
g  x   cx
2ax  cx  ax2  b  c 2acx2  acx2  bc
Applying the quotient d
z x  


formula:
dx
c2 x2
c2 x2
acx2  bc ax2  b


2 2
c x
cx2
Note that at x=0 neither the primitive nor the marginal function are defined!
Chain Rule
We need chain rule when we want to differentiate a function of one variable that is
in turn a function of another variable such as
z  f  y   f g x  zx
Function z is called a composite function since in fact it is a combination of two other
functions.
The chain rule:
Examples:
dz dz dy d
d


f y
g x   f  y g x 
dx dy dx dy
dx
z  3 y 2 , y  2x  5


dz
d
d
2 x  5  6 y  2  y  2x  5  62x  5 2  122x  5

3y2
dx dy
dx
Suppose a firm’s total revenue function, TR(Q)=f(Q), while the output Q itself is a
function of labor: Q=g(L).
Additional revenue due to one more worker is
dTR
d
d

f Q 
g L   f Q g L  equal to the additional output produced by the
dL
dQ
dL
worker times the money at which additional
output is sold.
Application 1: a Simple Market Model
Q  a  bP

Q  c  dp
Q
Demand; a>0, b>0
Supply; c>0, d>0
S
a
What happens to the equilibrium
quantity Q*when parameter a
increases?
?
a
Q
Equilibrium Q and P
ac
 *
P



bd

Q *  a d  b c

bd

Q*
d

 0  Q* 
a b  d
*
D
P
*
The equilibrium price level
increases as well since
P
P *
1

0
a
bd
P*
Application 1: a Simple Market Model
Consider an increase in the demand
parameter b:
Q
S
Q*  cb  d   ad  bc1
ad  bc



b
b  d 2
b  d 2
Q*
Q
D
*
Q  a  bP
D
P
Q *
0
b