252 Chapter 9Functions of several variables
and each of these can be differentiated with respect to either variable to give four
partial second derivatives. Differentiation of∂z 2 ∂xgives
and differentiation of∂z 2 ∂ygives
For a function of two variables, there are a possible 8 partial third derivatives, 16
fourth derivatives, and so on; in general a possible 2
nnth derivatives. In terms of
Figure 9.3, the first derivative∂z 2 ∂xis the gradient at a point, P say, on the curve
APB, and∂
2z 2 ∂x
2is the rate of change of this gradient as the point P moves along
the curve. On the other hand, the ‘mixed’ second derivative∂
2z 2 ∂y∂z 1 = 1 ∂(∂z 2 ∂x) 2 ∂y
is the rate of change of the gradient∂z 2 ∂x(in the x-direction) as the point P moves
along the curve DPE (in the perpendicular y-direction).
We note that, for the cubic function, the two mixed second derivatives are identical.
This is true for functions whose first derivatives are continuous, and is therefore (very
nearly) always true in practice:
(9.6)
Similar results are obtained for higher derivatives.
Alternative notations
The above symbols for partial derivatives become unwieldy for the higher derivatives,
and the following more compact notation is often used:
(9.7)
In this notation, equation (9.6) becomesf
xy1 = 1 f
yxand, for example,f
xxy1 = 1 f
xyx1 = 1 f
yxx(subject to the relevant continuity conditions).
In some applications, particularly in thermodynamics, it is necessary to specify
explicitly which variables (or combinations of variables) are to be kept constant. This
is achieved by adding the constant variables as subscripts to the ordinary symbol for the
partial derivative. For example, for a function of three variablesf(x, y, z), the symbol
(9.8)
means the derivative offwith respect to xat constantyandz.
yz
f
x
,
∂
∂
f
f
x
f
f
x
f
f
xy
f
f
xy
x xx xy xyz=
∂
∂
,=
∂
∂
,=
∂
∂∂
=
∂
∂∂
2223,
∂∂
,
z
...
∂
∂∂
=
∂
∂∂
22z
xy
z
yx
∂
∂
∂
∂
=
∂
∂
=+ ,
∂
∂
∂
∂
=
∂
y
z
y
z
y
xy
x
z
y
22624
2246
z
xy
xy
∂∂
=+
∂
∂
∂
∂
=
∂
∂
=+,
∂
∂
∂
∂
=
∂
x
z
x
z
x
xy
y
z
x
22264
zz
yx
xy
∂∂
=+ 46