Definition of the partial derivative

5
Partial differentiation
In chapter 2, we discussed functions f of only one variable x, which were usually
written f(x). Certain constants and parameters may also have appeared in the
definition of f, e.g. f(x) = ax + 2 contains the constant 2 and the parameter a, but
only x was considered as a variable and only the derivatives f (n) (x) = dn f/dxn
were defined.
However, we may equally well consider functions that depend on more than one
variable, e.g. the function f(x, y) = x2 + 3xy, which depends on the two variables
x and y. For any pair of values x, y, the function f(x, y) has a well-defined value,
e.g. f(2, 3) = 22. This notion can clearly be extended to functions dependent on
more than two variables. For the n-variable case, we write f(x1 , x2 , . . . , xn ) for
a function that depends on the variables x1 , x2 , . . . , xn . When n = 2, x1 and x2
correspond to the variables x and y used above.
Functions of one variable, like f(x), can be represented by a graph on a
plane sheet of paper, and it is apparent that functions of two variables can,
with little effort, be represented by a surface in three-dimensional space. Thus,
we may also picture f(x, y) as describing the variation of height with position
in a mountainous landscape. Functions of many variables, however, are usually
very difficult to visualise and so the preliminary discussion in this chapter will
concentrate on functions of just two variables.
5.1 Definition of the partial derivative
It is clear that a function f(x, y) of two variables will have a gradient in all
directions in the xy-plane. A general expression for this rate of change can be
found and will be discussed in the next section. However, we first consider the
simpler case of finding the rate of change of f(x, y) in the positive x- and ydirections. These rates of change are called the partial derivatives with respect
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PARTIAL DIFFERENTIATION
to x and y respectively, and they are extremely important in a wide range of
physical applications.
For a function of two variables f(x, y) we may define the derivative with respect
to x, for example, by saying that it is that for a one-variable function when y is
held fixed and treated as a constant. To signify that a derivative is with respect
to x, but at the same time to recognize that a derivative with respect to y also
exists, the former is denoted by ∂f/∂x and is the partial derivative of f(x, y) with
respect to x. Similarly, the partial derivative of f with respect to y is denoted by
∂f/∂y.
To define formally the partial derivative of f(x, y) with respect to x, we have
f(x + ∆x, y) − f(x, y)
∂f
= lim
,
(5.1)
∂x ∆x→0
∆x
provided that the limit exists. This is much the same as for the derivative of a
one-variable function. The other partial derivative of f(x, y) is similarly defined
as a limit (provided it exists):
f(x, y + ∆y) − f(x, y)
∂f
= lim
.
∂y ∆y→0
∆y
(5.2)
It is common practice in connection with partial derivatives of functions
involving more than one variable to indicate those variables that are held constant
by writing them as subscripts to the derivative symbol. Thus, the partial derivatives
defined in (5.1) and (5.2) would be written respectively as
∂f
∂f
and
.
∂x y
∂y x
In this form, the subscript shows explicitly which variable is to be kept constant.
A more compact notation for these partial derivatives is fx and fy . However, it is
extremely important when using partial derivatives to remember which variables
are being held constant and it is wise to write out the partial derivative in explicit
form if there is any possibility of confusion.
The extension of the definitions (5.1), (5.2) to the general n-variable case is
straightforward and can be written formally as
[f(x1 , x2 , . . . , xi + ∆xi , . . . , xn ) − f(x1 , x2 , . . . , xi , . . . , xn )]
∂f(x1 , x2 , . . . , xn )
= lim
,
∆xi →0
∂xi
∆xi
provided that the limit exists.
Just as for one-variable functions, second (and higher) partial derivatives may
be defined in a similar way. For a two-variable function f(x, y) they are
∂ ∂f
∂2 f
∂2 f
∂ ∂f
= 2 = fxx ,
= 2 = fyy ,
∂x ∂x
∂x
∂y ∂y
∂y
2
∂ ∂f
∂ ∂f
∂ f
∂2 f
= fxy ,
= fyx .
=
=
∂x ∂y
∂x∂y
∂y ∂x
∂y∂x
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