Differential Calculus

Differential Calculus

Partial Differentiation
(Partial Differential Coefficient)
Prepared by:
Dr. Sunil
NIT Hamirpur (HP)
(Last updated on 01-08-2009)
Latest update available at: http://www.freewebs.com/sunilnit/




(37 Solved problems and 00 Home assignments)

Introduction
Partial differentiation is the process of finding partial derivatives. A partial
derivative of several variables is the ordinary derivative with respect to one of the
variables when all the remaining variables are held constant. All the rules of
differentiation applicable to function of a single independent variable are also
applicable in partial differentiation with the only difference that while differentiating
(partially) with respect to one variable, all the other variables are treated
(temporarily) as constants.


Differential Coefficient:
If y is a function of only one independent variable, say x, then we can write
y = f(x).
Then, the rate of change of y w.r.t. x i.e. the derivative of y w.r.t. x is defined as
dy
 Lim
y
 Lim
y  y   y  Lim f x  x   f ( x )
dx x  0 x x  0 x x  0 x
where y is the change or increment of y corresponding to the increment x of the
independent variable x.

,Partial Differentiation: Partial Differential Coefficient Prepared by: Dr. Sunil, NIT Hamirpur 2


Partial Differential Coefficient:
Let u be a function of x and y i.e. u = f(x, y).
Then the partial differential coefficient of u (i.e. f(x, y) w.r.t. x (keeping y as constant) is
defined and written as
u f x  x , y   f ( x , y) f
 Lim  u x  fx  .
x x  0 x x
Similarly, the partial differential coefficient of u (i.e. f(x, y) w.r.t. y (keeping x as
constant) is defined and written as
u f x , y  y   f ( x , y) f
 Lim  uy  fy  .
y y 0 y y
Similarly, we can find

 2u   u   2 u   u   2 u   u   2 u   u 
2
   , 2
 
 
 ,    ,   .
x x  x  y y  y  xy x  y  yx y  x 

 2u  2u
Also, it can be verified that  .
xy yx
Notation:
u f
The partial derivative is also denoted by or f x ( x , y, z) or fx or Dxf or
x x
f1 (x, y, z) , where the subscripts x and 1 denote the variable w.r.t. x which the partial
differentiation is carried out.
u f
Thus, we can have   f y x, y, z   f y  D y f  f 2 x, y, z  etc.
y y
The value of a partial derivative at a point (a, b, c) is denoted by
u u
  f x a , b, c  .
x x a , y  b , z c x a ,b,c 

,Partial Differentiation: Partial Differential Coefficient Prepared by: Dr. Sunil, NIT Hamirpur 3


Geometrical Interpretation of partial derivatives:
(Geometrical interpretation of a partial derivative of a function of two variables)
z  f ( x , y) represents the equation of surface in xyz-coordinate system. Let APB
be the curve, which is drawn on a plane through any point P on the surface parallel to the
xz-plane.
As point P moves along the curve APB, its coordinates z and x vary while y remains
constant. The slope of the tangent line at P to APB represents the ‘rate at which z changes
w.r.t. x’.
z-axis B z-axis D
P P


A C

x-axis y-axis
O O


y-axis x-axis

Figure 1 Figure 2
z
Thus  tan  = slope of the curve APB at the point P (see fig.1).
x
z
Similarly,  tan  = slope of the curve CPD at the point P (see fig.2).
y
Higher Order Parallel Derivatives:
Partial derivatives of higher order, of a function f(x, y, z) are calculated by
successive differentiate. Thus, if u = f(x, y, z) then

 2u  2f   f   2u  2f   f 
     f xx  f11 ,      f yx  f 21 ,
x 2 x 2 x  x  xy xy x  y 

 2u  2f   f   2u  2f   f 
    xy  f  f 12 ,      f yy  f 22 ,
yx yx y  x  y 2 y 2 y  y 

 3u    2 f      f 
      f yzz  f 233 ,
z 2 y z  zy  z  z  y 

 4u    3 f       2 f 
     f zzyx  f 3321 .
xyz 2 x  yz 2  x  y  z 2 

, Partial Differentiation: Partial Differential Coefficient Prepared by: Dr. Sunil, NIT Hamirpur 4


f
The partial derivative obtained by differentiating once in known as first order partial
x

 2f  2f  2f  2f
derivative, while , , , which are obtained by differentiating twice are
x 2 y 2 xy yx

known as second order derivatives. 3rd order, 4th order derivatives involve 3, 4, times
differentiation respectively.

 2f  2f
Note 1: The crossed or mixed partial derivatives and are, in general, equal
yx xy

 2f  2f
 .
yx xy
i.e. the order of differentiation is immaterial if the derivatives involved are continuous.
Note 2: In the subscript notation, the subscript are written in the same order in which
differentiation is carried out, while in '' notation the order is opposite, for example

 2u   u 
    f xy .
yx y  x 
Note 3: A function of 2 variables has two first order derivatives, four second order
derivatives and 2nd of nth order derivatives. A function of m independent variables will have
mn derivatives of order n.


Now let us solve some problems related to the above-mentioned topics:

y  2u  2u
Q.No.1.: If u  tan 1  , then prove that   0.
x x 2 y 2

y
Sol.: Here u  tan 1  .
x
u
Since  the p. d. coefficient of u w. r. t. x (keeping y as constant)
x
1  y  y
  2   2 .
y  x  x  y2
2
1
x2


 2u
    

  u     y  x 2  y 2 .0  2x  y 

 2 xy
....(i)
x 2  2 2
x  x  x  x  y 
x 2  y2
2
 
x 2  y2  
2