Lecture 9 - Partial Differentiations of Functions of Two or more Variables | Calculus - Engineering Mathematics PDF Download

 Page 1


 Partial Differentiations of Functions of Two or more Variables 
 
Institute of Lifelong Learning, University of Delhi                                                       
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Lesson: Partial Differentiations of Functions of Two or more 
Variables 
Lesson Developer: Kapil Kumar 
Department/College: Assistant Professor, Department of 
Mathematics, A.R.S.D. College, University of Delhi 
 
 
 
 
 
 
 
 
 
 
Page 2


 Partial Differentiations of Functions of Two or more Variables 
 
Institute of Lifelong Learning, University of Delhi                                                       
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Lesson: Partial Differentiations of Functions of Two or more 
Variables 
Lesson Developer: Kapil Kumar 
Department/College: Assistant Professor, Department of 
Mathematics, A.R.S.D. College, University of Delhi 
 
 
 
 
 
 
 
 
 
 
 Partial Differentiations of Functions of Two or more Variables 
 
Institute of Lifelong Learning, University of Delhi                                                       
 
Table of Contents: 
 Chapter: Partial Differentiations of Functions of Two or more 
Variables 
? 1: Learning outcomes 
? 2: Introduction 
? 3: Partial Differentiation of a Function of Two Variable 
o 3.1. Partial Derivative of ( , ) f x y at a Point 
00
( , ) xy 
o 3.2. Geometrical Interpretation of Partial Differentiation of 
First Order 
? 4: Partial Derivatives and Continuity 
? 5: Partial Derivatives of Higher Order 
? 6: Change in the Order of Partial Differentiation 
? 7: Homogeneous Function 
? 8: Euler’s Theorem for Partial Derivatives of a Homogeneous 
Function 
? Exercises 
? Summary 
? References 
 
1. Learning outcomes: 
After studying this chapter you should be able to understand the 
? Partial Differentiation of a Function of Two Variable 
? Partial Derivatives and Continuity 
? Partial Derivatives of Higher Order 
? Partial Derivatives of Higher Order 
? Homogeneous Function 
? Euler’s Theorem for Partial Derivatives of a Homogeneous Function 
 
 
 
 
 
Page 3


 Partial Differentiations of Functions of Two or more Variables 
 
Institute of Lifelong Learning, University of Delhi                                                       
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Lesson: Partial Differentiations of Functions of Two or more 
Variables 
Lesson Developer: Kapil Kumar 
Department/College: Assistant Professor, Department of 
Mathematics, A.R.S.D. College, University of Delhi 
 
 
 
 
 
 
 
 
 
 
 Partial Differentiations of Functions of Two or more Variables 
 
Institute of Lifelong Learning, University of Delhi                                                       
 
Table of Contents: 
 Chapter: Partial Differentiations of Functions of Two or more 
Variables 
? 1: Learning outcomes 
? 2: Introduction 
? 3: Partial Differentiation of a Function of Two Variable 
o 3.1. Partial Derivative of ( , ) f x y at a Point 
00
( , ) xy 
o 3.2. Geometrical Interpretation of Partial Differentiation of 
First Order 
? 4: Partial Derivatives and Continuity 
? 5: Partial Derivatives of Higher Order 
? 6: Change in the Order of Partial Differentiation 
? 7: Homogeneous Function 
? 8: Euler’s Theorem for Partial Derivatives of a Homogeneous 
Function 
? Exercises 
? Summary 
? References 
 
1. Learning outcomes: 
After studying this chapter you should be able to understand the 
? Partial Differentiation of a Function of Two Variable 
? Partial Derivatives and Continuity 
? Partial Derivatives of Higher Order 
? Partial Derivatives of Higher Order 
? Homogeneous Function 
? Euler’s Theorem for Partial Derivatives of a Homogeneous Function 
 
 
 
 
 
 Partial Differentiations of Functions of Two or more Variables 
 
Institute of Lifelong Learning, University of Delhi                                                       
 
2. Introduction: 
The calculus of several variables is basically single-variable calculus applied 
to several variables one at a time. The ordinary derivative of a functions of 
several variables keeping all other independent variables constant is called 
the partial derivative of the function with respect to the variable. In this 
lesson, we will study how the partial derivatives are defined and interpreted 
and how the partial derivatives of a function are calculated by applying the 
rules for differentiating functions of a single variable. 
3. Partial Differentiation of a Function of Two Variable: 
Partial derivative of function of two variables ( , ) f x y with respect to x is 
denoted by ( , )
xx
f
or f or f x y
x
?
?
 and defined as 
  
0
( , ) ( , )
lim
h
f f x h y f x y
xh
?
? ? ?
?
?
 
provided the limit exist. 
Similarly the partial derivative of function ( , ) f x y with respect to y is 
denoted by ( , )
yy
f
or f or f x y
y
?
?
 and defined as 
  
0
( , ) ( , )
lim
k
f f x y k f x y
yk
?
? ? ?
?
?
 
provided the limit exist. 
3.1. Partial Derivative of ( , ) f x y at a Point 
00
( , ) xy :  
Partial derivatives of ( , ) f x y with respect to x at a particular point 
00
( , ) xy are 
often denoted by 
 
00
00
( , )
( , )
x
xy
f
or f x y
x
? ??
??
?
??
  
and defined as 
 
00
0 0 0 0
00
0
( , )
( , ) ( , )
( , ) lim
x
h
xy
f x h y f x y f
f x y
xh
?
?? ? ??
??
??
?
??
 
Page 4


 Partial Differentiations of Functions of Two or more Variables 
 
Institute of Lifelong Learning, University of Delhi                                                       
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Lesson: Partial Differentiations of Functions of Two or more 
Variables 
Lesson Developer: Kapil Kumar 
Department/College: Assistant Professor, Department of 
Mathematics, A.R.S.D. College, University of Delhi 
 
 
 
 
 
 
 
 
 
 
 Partial Differentiations of Functions of Two or more Variables 
 
Institute of Lifelong Learning, University of Delhi                                                       
 
Table of Contents: 
 Chapter: Partial Differentiations of Functions of Two or more 
Variables 
? 1: Learning outcomes 
? 2: Introduction 
? 3: Partial Differentiation of a Function of Two Variable 
o 3.1. Partial Derivative of ( , ) f x y at a Point 
00
( , ) xy 
o 3.2. Geometrical Interpretation of Partial Differentiation of 
First Order 
? 4: Partial Derivatives and Continuity 
? 5: Partial Derivatives of Higher Order 
? 6: Change in the Order of Partial Differentiation 
? 7: Homogeneous Function 
? 8: Euler’s Theorem for Partial Derivatives of a Homogeneous 
Function 
? Exercises 
? Summary 
? References 
 
1. Learning outcomes: 
After studying this chapter you should be able to understand the 
? Partial Differentiation of a Function of Two Variable 
? Partial Derivatives and Continuity 
? Partial Derivatives of Higher Order 
? Partial Derivatives of Higher Order 
? Homogeneous Function 
? Euler’s Theorem for Partial Derivatives of a Homogeneous Function 
 
 
 
 
 
 Partial Differentiations of Functions of Two or more Variables 
 
Institute of Lifelong Learning, University of Delhi                                                       
 
2. Introduction: 
The calculus of several variables is basically single-variable calculus applied 
to several variables one at a time. The ordinary derivative of a functions of 
several variables keeping all other independent variables constant is called 
the partial derivative of the function with respect to the variable. In this 
lesson, we will study how the partial derivatives are defined and interpreted 
and how the partial derivatives of a function are calculated by applying the 
rules for differentiating functions of a single variable. 
3. Partial Differentiation of a Function of Two Variable: 
Partial derivative of function of two variables ( , ) f x y with respect to x is 
denoted by ( , )
xx
f
or f or f x y
x
?
?
 and defined as 
  
0
( , ) ( , )
lim
h
f f x h y f x y
xh
?
? ? ?
?
?
 
provided the limit exist. 
Similarly the partial derivative of function ( , ) f x y with respect to y is 
denoted by ( , )
yy
f
or f or f x y
y
?
?
 and defined as 
  
0
( , ) ( , )
lim
k
f f x y k f x y
yk
?
? ? ?
?
?
 
provided the limit exist. 
3.1. Partial Derivative of ( , ) f x y at a Point 
00
( , ) xy :  
Partial derivatives of ( , ) f x y with respect to x at a particular point 
00
( , ) xy are 
often denoted by 
 
00
00
( , )
( , )
x
xy
f
or f x y
x
? ??
??
?
??
  
and defined as 
 
00
0 0 0 0
00
0
( , )
( , ) ( , )
( , ) lim
x
h
xy
f x h y f x y f
f x y
xh
?
?? ? ??
??
??
?
??
 
 Partial Differentiations of Functions of Two or more Variables 
 
Institute of Lifelong Learning, University of Delhi                                                       
 
provided the limit exists. 
Similarly, the partial derivatives of ( , ) f x y with respect to y at a particular 
point 
00
( , ) xy are often denoted by 
 
00
00
( , )
( , )
y
xy
f
or f x y
y
?? ?
??
?
??
  
and defined as 
 
00
0 0 0 0
00
0
( , )
( , k) ( , )
( , ) lim
y
k
xy
f x y f x y f
f x y
yk
?
?? ?? ?
??
??
?
??
 
provided the limit exists. 
3.2. Geometrical Interpretation of Partial Differentiation of First 
Order: 
Let ( , ) z f x y ? represents a surface geometrically and let 
0 0 0 0
[( , ), ( , )] P x y f x y 
be a point on the surface corresponding to the point 
00
( , ) xy of the domain of 
the function. 
If a variable point, starting from P changes, it position on the surface such 
that y remains constantly equal to b, then it is clear that the locus of the 
point is the curve of intersection of the surface and the plane yb ? . 
On the curve x and z vary according to the relation ( , ) z f x b ? . Also 
00
( , ) xy
z
x
? ??
??
?
??
 
is the ordinary derivative of ( , ) f x b w.r.t. x for xa ? . Hence, we see that 
00
( , ) xy
z
x
? ??
??
?
??
 denotes the tangent of the angle which the tangent to the curve, in 
which the plane 
0
yy ? parallel to the ZX plane cuts the surface at 
0 0 0 0
[( , ), ( , )] P x y f x y makes with x-axis. 
 Similarly it may be seen that 
00
( , ) xy
z
y
?? ?
??
?
??
 denotes the tangent of the 
angle which the tangent at 
0 0 0 0
[( , ), ( , )] P x y f x y to the curve of intersection of 
the surface and the plane xa ? parallel to the ZY plane makes with y-axis. 
Page 5


 Partial Differentiations of Functions of Two or more Variables 
 
Institute of Lifelong Learning, University of Delhi                                                       
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Lesson: Partial Differentiations of Functions of Two or more 
Variables 
Lesson Developer: Kapil Kumar 
Department/College: Assistant Professor, Department of 
Mathematics, A.R.S.D. College, University of Delhi 
 
 
 
 
 
 
 
 
 
 
 Partial Differentiations of Functions of Two or more Variables 
 
Institute of Lifelong Learning, University of Delhi                                                       
 
Table of Contents: 
 Chapter: Partial Differentiations of Functions of Two or more 
Variables 
? 1: Learning outcomes 
? 2: Introduction 
? 3: Partial Differentiation of a Function of Two Variable 
o 3.1. Partial Derivative of ( , ) f x y at a Point 
00
( , ) xy 
o 3.2. Geometrical Interpretation of Partial Differentiation of 
First Order 
? 4: Partial Derivatives and Continuity 
? 5: Partial Derivatives of Higher Order 
? 6: Change in the Order of Partial Differentiation 
? 7: Homogeneous Function 
? 8: Euler’s Theorem for Partial Derivatives of a Homogeneous 
Function 
? Exercises 
? Summary 
? References 
 
1. Learning outcomes: 
After studying this chapter you should be able to understand the 
? Partial Differentiation of a Function of Two Variable 
? Partial Derivatives and Continuity 
? Partial Derivatives of Higher Order 
? Partial Derivatives of Higher Order 
? Homogeneous Function 
? Euler’s Theorem for Partial Derivatives of a Homogeneous Function 
 
 
 
 
 
 Partial Differentiations of Functions of Two or more Variables 
 
Institute of Lifelong Learning, University of Delhi                                                       
 
2. Introduction: 
The calculus of several variables is basically single-variable calculus applied 
to several variables one at a time. The ordinary derivative of a functions of 
several variables keeping all other independent variables constant is called 
the partial derivative of the function with respect to the variable. In this 
lesson, we will study how the partial derivatives are defined and interpreted 
and how the partial derivatives of a function are calculated by applying the 
rules for differentiating functions of a single variable. 
3. Partial Differentiation of a Function of Two Variable: 
Partial derivative of function of two variables ( , ) f x y with respect to x is 
denoted by ( , )
xx
f
or f or f x y
x
?
?
 and defined as 
  
0
( , ) ( , )
lim
h
f f x h y f x y
xh
?
? ? ?
?
?
 
provided the limit exist. 
Similarly the partial derivative of function ( , ) f x y with respect to y is 
denoted by ( , )
yy
f
or f or f x y
y
?
?
 and defined as 
  
0
( , ) ( , )
lim
k
f f x y k f x y
yk
?
? ? ?
?
?
 
provided the limit exist. 
3.1. Partial Derivative of ( , ) f x y at a Point 
00
( , ) xy :  
Partial derivatives of ( , ) f x y with respect to x at a particular point 
00
( , ) xy are 
often denoted by 
 
00
00
( , )
( , )
x
xy
f
or f x y
x
? ??
??
?
??
  
and defined as 
 
00
0 0 0 0
00
0
( , )
( , ) ( , )
( , ) lim
x
h
xy
f x h y f x y f
f x y
xh
?
?? ? ??
??
??
?
??
 
 Partial Differentiations of Functions of Two or more Variables 
 
Institute of Lifelong Learning, University of Delhi                                                       
 
provided the limit exists. 
Similarly, the partial derivatives of ( , ) f x y with respect to y at a particular 
point 
00
( , ) xy are often denoted by 
 
00
00
( , )
( , )
y
xy
f
or f x y
y
?? ?
??
?
??
  
and defined as 
 
00
0 0 0 0
00
0
( , )
( , k) ( , )
( , ) lim
y
k
xy
f x y f x y f
f x y
yk
?
?? ?? ?
??
??
?
??
 
provided the limit exists. 
3.2. Geometrical Interpretation of Partial Differentiation of First 
Order: 
Let ( , ) z f x y ? represents a surface geometrically and let 
0 0 0 0
[( , ), ( , )] P x y f x y 
be a point on the surface corresponding to the point 
00
( , ) xy of the domain of 
the function. 
If a variable point, starting from P changes, it position on the surface such 
that y remains constantly equal to b, then it is clear that the locus of the 
point is the curve of intersection of the surface and the plane yb ? . 
On the curve x and z vary according to the relation ( , ) z f x b ? . Also 
00
( , ) xy
z
x
? ??
??
?
??
 
is the ordinary derivative of ( , ) f x b w.r.t. x for xa ? . Hence, we see that 
00
( , ) xy
z
x
? ??
??
?
??
 denotes the tangent of the angle which the tangent to the curve, in 
which the plane 
0
yy ? parallel to the ZX plane cuts the surface at 
0 0 0 0
[( , ), ( , )] P x y f x y makes with x-axis. 
 Similarly it may be seen that 
00
( , ) xy
z
y
?? ?
??
?
??
 denotes the tangent of the 
angle which the tangent at 
0 0 0 0
[( , ), ( , )] P x y f x y to the curve of intersection of 
the surface and the plane xa ? parallel to the ZY plane makes with y-axis. 
 Partial Differentiations of Functions of Two or more Variables 
 
Institute of Lifelong Learning, University of Delhi                                                       
 
 The slope of the curve 
0
( , ) z f x y ? at the point 
0 0 0 0
[( , ), ( , )] P x y f x y in the 
plane 
0
yy ? is the value of the partial derivative of f with respect to x at 
00
( , ) xy . The tangent line to the curve at P is the line in the plane 
0
yy ? that 
passes through P with this slope. The partial derivative 
f
x
? ??
??
?
??
 at 
00
( , ) xy gives 
the rate of change of f with respect to x when y is held fixed at the value 
0
y . 
This is the rate of change of f in the direction of i at 
00
( , ) xy . 
Value Addition: Note 
The definition of 
f
x
?
?
 and 
f
y
?
?
 give us two different ways of differentiating the 
function ( , ) f x y with respect to x in the usual way while treating y as a 
constant and with respect to y in the usual way while treating x as constant. 
 
Example 1: If 
2
3
( , )
xy
f x y x y e ?? , find 
x
f and 
y
f . 
Solution: We have 
 
2
3
( , )
xy
f x y x y e ?? 
(I) Treating y as a constant and differentiation w.r.t. x, we have 
 
? ?
2
3
( , )
xy
f
f x y x y e
x x x
? ? ?
? ? ?
? ? ?
 
 
? ? ? ?
2
3 xy
x y e
xx
??
??
??
 
 
2
22
3
xy
x y y e ?? 
(II) Treating x as a constant and differentiation w.r.t. y, we have 
 
? ?
2
3
( , )
xy
f
f x y x y e
y y y
? ? ?
? ? ?
? ? ?
 
 
? ? ? ?
2
3 xy
x y e
yy
??
??
??
 
 
2
3
2
xy
x xye ??