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UNIT 3 PARTIAL DIFFERENTIATION
Structure
3.0 Objectives
3.1 Introduction
3.2 Concept of Partial Derivative
3.2.1 Partial Derivative Defined
3.2.2 Higher Order Partial Derivatives
3.2.3 Cross- Partial Derivative
3.3 Total Differential
3.3.1 Total Derivative
3.4 Differentiation of Function of Functions
3.4.1 Derivatives of Implicit Functions
3.5 Application in Economics
3.6 Homogeneous Functions and Their Properties
3.7 Let Us Sum Up
3.8 Key Words
3.9 Some Useful Books
3.10 Answers or Hints to Check Your Progress
3.0 OBJECTIVES
After going through this unit, you will be able to:
• extend the differentiation technique from single variable to multivariate
cases; and
• include some new concept of derivative associated with will be
introduced. Then we will discuss the properties of some special functions.
3.1 INTRODUCTION
So far, we have considered the functions of a single independent variable.
However, most of the economic problems deal with the multivariate cases.
For example, utility of a consumer depends on the consumption of multiple
commodities and the production of a commodity depends on factors like land,
labor, and capital and so on.
Like this, other examples can be included to capture more interesting
economic problems. Therefore, there is a need to analyse the concept of
derivative for the class of functions with more than one independent variables.
3.2 CONCEPT OF PARTIAL DERIVATIVE
For simplicity, we shall start with function of two independent variables. Let
the function be y = f(x , x ). When we take such a function y = f(x , x ), with
1 2 1 2
two independent variables, change in y can be examined as result of a change
in either x or x or by both by assuming these to be independent. Thus, if we
1 2,
change only x , it will not affect x but only y. The rate of change in y,
1 2
assuming x2 to remain constant, can be obtained by finding partial derivative
46
of y with respect to x . Similarly, if x is changed alone, the rate of change in Partial Differentiation
1 2
y can be obtained from the partial derivative of y with respect to x .
2
Proceeding one-step further and assuming that x and x are related to each
1 2
other, if we change x , there will be change in x (even when x is not changed
1 2 2
exogenously), as a result of the change in x1. Thus, whenever we change the
dependent variable, y is affected in two ways: first, directly by the change in
x andsecondly, indirectly by the change in x via the change in x . The rate of
1 2 1
change of y due to a change in x1 in this case is measured by total derivative
of y with respect to x1.
3.2.1 Partial Derivative Defined
Partial derivative is referred to the derivative of a multivariate function when
only one of the independent variables is allowed to change, other variables
remaining constant.
As an example, consider the bivariate function,
U = f(x , x ).
1 2
If we let only x to change, x remaining constant, the rate of change of U with
1 2
respect to x is called the partial derivative of U with respect to x . Let the
1 1
change in x is ∆x , x remaining constant. The corresponding change in U is
1 1 2
∆U. Then the difference quotient is given by
00
∆U f (,xx+∆ x)−f(x,x)
11212
=
∆+xx()∆x−x
1111
If we try to obtain the limiting value of this difference quotient as ∆x Æ0, we
1
get the partial derivative of U with respect to x1. This is denoted by
∂∂Uf
, or simply f .
∂∂x x 1
11
lim lim 00
∆U f (,xx+∆ x)−f(x,x)
11212
Thus, f ==
1 ∆→xx00∆→
∆∆xx
11
11
Similarly,
lim lim 00
∆U f (,xx+∆x)−f(xx,)
12 2 12
f2 ==
∆→xx00∆→
∆∆xx
22
22
Example: Find the partial derivative of the function
f(x x ), = x 3 + 2x 2 x + 3 x x + 4x 2.
1, 2 1 1 2 1 2 2
To obtain f we have to differentiate the above function with respect to x ,
1 1
treating x as constant.
2
Thus, f = 3x 2 + 4x x + 3x
1 1 1 2 2
Similarly, f = 2x 2 + 3x + 8x
2 1 1 2
Note that these partial derivatives are functions of the some dependent
variables as in case of the original function f.
we write, If y = f(x x ),
1, 2
then f =f (x , x ), f =f(x , x ).
1 1 1 2 2 1 2
This is because first derivative at any point was not only dependent on the
value of the independent variable, but also on the other variables, which are 47
Introduction to kept constant. First order partial derivative gives us the marginal values. For
Differential Calculus example, if the function y=f(x , x ) defines the total utility of the commodities
1 2
x , x then ∂U , gives us additional utility that can be obtained from the use
1 2, x
∂ 1
of one additional unit of x . Thus, ∂U is the marginal utility of x (i =1, 2).
1 ∂x i
i
3.2.2 Higher Order Partial Derivatives
We have already seen, if y = f(x , x ), then f(x , x ) (for i=1, 2) gives the
1 2 i 1 2
partial derivatives.
So long as this is true, we can repeat the process of partial differentiation and
get higher order partial derivatives. However, once the first order partial
derivative ceases to be a function of some choice variables as the primitive
function, higher order partial derivatives are no longer obtainable.
Example:
3 2
i) Differentiate z = 6x +5x +10xy partially with respect to x twice.
Ans:
z = 18x2 + 10x + 10y
x
Zxx = 36x + 10
ii) Find f11, f22 for the function
f xx12, = ex +x + 3x x
( ) 1 2 1 2
Ans:
x +x
f = e + 3x
1 1 2 2
x +x
f = e + 3x
2 1 2 1
Thus,
x +x
f = e + 3
11 1 2
x +x
f = e + 3
22 1 2
3.2.3 Cross- Partial Derivatives
Consider the function f(x , y ). The partial derivative of the function gives f
1 2 1
andf (defined above) The cross–partial derivatives are defined as follows:
2 .
f = ∂f1 And f = ∂f2
12 ∂x 21 ∂x
2 1
By Young’s theorem, f12 = f21.
More generally, for a n-variable function y = f(x , ---, x ).
1 n
f = f (for all i, j = 1, 2,-------, n)
ji ij
Note that such a condition holds when both the partial derivatives exist and
one of them is continuous.
48
Example: Partial Differentiation
3 5 2 4
i) f xy, = x + y + 4x y + 2 xy, find f and f .
( ) xy yx
2 4
fx = 3x + 8xy + 2y
4 2 3
fy = 5y + 16x y + 2 x
f = 32x y3 + 2 = f
xy yx
ii) f xx12, = log (x 2 + x 2). Find the second order partial derivatives.
( ) 1 2
2x 2x
f = 1 , f = 2
1 x 2 + x 2 2 x 2 + x 2
1 2 1 2
2(x 2 − x 2)
f = 2 1
11 (x 2 + x 2)2
1 2
4xx
ff== 12
12 21 222
()xx+
12
2(x 2 − x 2)
f = 1 2
22 (x 2 + x 2)2
1 2
Check Your Progress 1
1) A production function is given by
1 2 2
Q = 2 L K
where, Q = level of output
L = Labor input employed
K = Capital input employed
Find the marginal productivities of labor and capital.
………………………………………………………………………………
………………………………………………………………………………
………………………………………………………………………………
………………………………………………………………………………
2) f xy, = x + y. Find f , f .
( ) xx yy
………………………………………………………………………………
………………………………………………………………………………
………………………………………………………………………………
……………………………………………………………………………… 49
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