Transcript Document

Differentiation
• Basic Rules of Differentiation
• The Product and Quotient Rules
• Marginal Functions in Economics
• Higher-Order Derivatives
Basic Differentiation Rules
d
1.
c  0
dx
 c is a constant 
Ex. f ( x)  5
f ( x)  0
 
d n
x  nx n1
2.
dx
Ex. f ( x)  x 7
f ( x)  7 x 6
 n is a real number 
Basic Differentiation Rules
d
d
3.
 cf ( x)   c  f ( x) 
dx
dx
 c is a constant 
8
f
(
x
)

3
x
Ex.
 
f ( x)  3 8 x7  24 x7
d
d
d
4. dx  f  x   g  x    dx  f ( x)  dx  g ( x) 
Ex. f ( x)  7  x12
f ( x)  0 12x11  12x11
More Differentiation Rules
5. Product Rule
d
d
d
 f  x   g  x     f ( x) g ( x)   g ( x)  f ( x)
dx
dx
dx



 1   x  2 x  5  21x
Ex. f ( x)  x  2 x  5 3x  8x  1


3
f ( x)  3x 2  2 3x7  8 x 2
Derivative
of the first
function
7
3
2
6
 16 x
Derivative of
the second
function
f ( x)  30x9  48x7 105x6  40x4  45x2  80x  2

More Differentiation Rules
6. Quotient Rule
d
d
g ( x)  f ( x)   f ( x)  g ( x) 


f
x
  
d
dx
dx


2
dx  g ( x) 
g
(
x
)
 
Sometimes remembered as:
d  hi  lo d  hi   hi d lo



dx  lo 
lo lo
More Differentiation Rules
6. Quotient Rule (cont.)
3x  5
Ex. f ( x)  2
x 2
Derivative of
the numerator
f ( x) 


Derivative of
the denominator

3 x 2  2  2 x  3x  5
x
2
2

3x2  10 x  6
x
2
2

2
2
More Differentiation Rules
The General Power Rule:
If h( x)   f ( x)
n
 n, real  then
h( x)  n  f ( x) 
n 1
 f ( x)

Ex. f ( x)  3x  4 x  3x  4 x
2

1
2

f ( x)  3 x  4 x
2
3x  2

3x 2  4 x

1 2
2

12
6x  4
7
2
x

1


Ex. G( x) 


 3x  5 
6
 2 x  1    3x  5 2   2 x  1 3 
G( x)  7 
 
2

 3x  5  
 3x  5 

 2x 1 
G( x)  7 

 3x  5 
6
13
 3x  5
2

91 2 x  1
 3x  5 
8
6
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.
Example
The monthly demand for T-shirts is given by
0  x  400
p  0.05x  25
where p denotes the wholesale unit price in dollars
and x denotes the quantity demanded. The monthly
cost function for these T-shirts is
C( x)  0.001x  2x  200
2
1. Find the revenue and profit functions.
2. Find the marginal cost, marginal revenue, and
marginal profit functions.
Solution
1. Find the revenue and profit functions.
Revenue = xp
 x  0.05x  25  0.05x  25x
2
Profit = revenue – cost

 0.05 x  25 x  0.001x  2 x  200
2
2
 0.049 x2  23x  200
2. Find the marginal cost, marginal revenue, and
marginal profit functions.
Marginal Cost = C ( x )
 0.002 x  2
......

Solution
2. (cont.) Find the marginal revenue and marginal
profit functions.
Marginal revenue = R( x )
 0.1x  25
Marginal profit = P( x )
 0.098 x  23
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
Second
Third
Fourth
nth
Notations
f 
d2y
dx 2
f 
d3y
dx3
(4)
d4y
dx 4
 n
dny
dx n
f
f
Example of Higher Derivatives
Given f ( x)  3x5  2 x3  14 find f ( x).
4
2

f ( x)  15x  6x
f ( x)  60x3 12x
2

f ( x)  180x 12
Example of Higher Derivatives
2x 1
Given f ( x) 
find f (2).
3x  2
f ( x) 
2  3 x  2   3  2 x  1
 3x  2 
f ( x)  14  3x  2
f (2) 
3
42
3(2)  2
3
2
3 

7
 3x  2 
42
 3x  2 
42 21
 3 
32
4
3
2
 7  3 x  2 
2