B.1 Derivatives
597
The definitions of the two partial derivatives for z = f (x, y) may now be
given. For y held fixed we have
∂z
∂x
≡
∂f
∂x
= lim
x→0
f (x + y) − f (x, y)
x
,
(B.1.4a)
while for x held fixed, we have
∂z
∂y
≡
∂f
∂y
= lim
y→0
f (x, y + − f (x, y)
y
.
(B.1.4b)
It is worth noting that although we may express the derivative f (x) of a
function f of the single variable x as the quotient of two differentials, df
and dx because the defining relation df (x 0 ) = f (x 0 )dx provides a 1 ↔ 1
relationship between the differential df at x 0 and the independent differential
dx for a given value x 0 of x, while from the corresponding relationship,
df (x 0 , y 0 ) =
∂f
∂x
y
(x 0 , y 0 )dx +
∂f
∂y
x
(x 0 , y 0 )dy ,
for a pair of given values (x 0 , y 0 ) of (x, y), it is no longer possible to
establish such 1 ↔ 1 relationships between the partial derivative functions
and the independent differentials df and dx or df and dy. For this reason,
partial derivatives must be treated as functions that cannot be obtained as the
quotients of dependent and independent differentials, and it makes no sense
to write ∂f , ∂x, or ∂y on its own.
Higher-order partial derivatives: Note from Eqs. (B.1.4a) and (B.1.4b), that
if z = f (x, y) is a function of the independent variables x and y, then the
partial derivatives
∂z
∂x
and
∂z
∂y
are also functions of x and y. Hence, these
(first) partial derivatives can be differentiated in turn with respect to x and y.
There are thus four second partial derivatives, namely,
∂ 2 z
∂x 2 ,
∂ 2 z
∂y 2 ,
∂ 2 z
∂x∂y
,
∂ 2 z
∂y∂x
;
(B.1.5a)
note that in general the mixed second partial derivatives are equal, i.e.,
∂ 2 z
∂x∂y
=
∂ 2 z
∂y∂x
.
(B.1.5b)
Chain rule for multivariate functions: Extension of the chain rule (B.1.2) to
functions of two variables is reasonably straightforward. If z is a function of
the two variables x and y, and x, y are in turn functions of the two variables
u and v, then the chain rule for the partial derivatives of z with respect to u
and v is
597
The definitions of the two partial derivatives for z = f (x, y) may now be
given. For y held fixed we have
∂z
∂x
≡
∂f
∂x
= lim
x→0
f (x + y) − f (x, y)
x
,
(B.1.4a)
while for x held fixed, we have
∂z
∂y
≡
∂f
∂y
= lim
y→0
f (x, y + − f (x, y)
y
.
(B.1.4b)
It is worth noting that although we may express the derivative f (x) of a
function f of the single variable x as the quotient of two differentials, df
and dx because the defining relation df (x 0 ) = f (x 0 )dx provides a 1 ↔ 1
relationship between the differential df at x 0 and the independent differential
dx for a given value x 0 of x, while from the corresponding relationship,
df (x 0 , y 0 ) =
∂f
∂x
y
(x 0 , y 0 )dx +
∂f
∂y
x
(x 0 , y 0 )dy ,
for a pair of given values (x 0 , y 0 ) of (x, y), it is no longer possible to
establish such 1 ↔ 1 relationships between the partial derivative functions
and the independent differentials df and dx or df and dy. For this reason,
partial derivatives must be treated as functions that cannot be obtained as the
quotients of dependent and independent differentials, and it makes no sense
to write ∂f , ∂x, or ∂y on its own.
Higher-order partial derivatives: Note from Eqs. (B.1.4a) and (B.1.4b), that
if z = f (x, y) is a function of the independent variables x and y, then the
partial derivatives
∂z
∂x
and
∂z
∂y
are also functions of x and y. Hence, these
(first) partial derivatives can be differentiated in turn with respect to x and y.
There are thus four second partial derivatives, namely,
∂ 2 z
∂x 2 ,
∂ 2 z
∂y 2 ,
∂ 2 z
∂x∂y
,
∂ 2 z
∂y∂x
;
(B.1.5a)
note that in general the mixed second partial derivatives are equal, i.e.,
∂ 2 z
∂x∂y
=
∂ 2 z
∂y∂x
.
(B.1.5b)
Chain rule for multivariate functions: Extension of the chain rule (B.1.2) to
functions of two variables is reasonably straightforward. If z is a function of
the two variables x and y, and x, y are in turn functions of the two variables
u and v, then the chain rule for the partial derivatives of z with respect to u
and v is