The total differential and total derivative

5.2 THE TOTAL DIFFERENTIAL AND TOTAL DERIVATIVE
Only three of the second derivatives are independent since the relation
∂2 f
∂2 f
=
,
∂x∂y
∂y∂x
is always obeyed, provided that the second partial derivatives are continuous
at the point in question. This relation often proves useful as a labour-saving
device when evaluating second partial derivatives. It can also be shown that for
a function of n variables, f(x1 , x2 , . . . , xn ), under the same conditions,
∂2 f
∂2 f
=
.
∂xi ∂xj
∂xj ∂xi
Find the first and second partial derivatives of the function
f(x, y) = 2x3 y 2 + y 3 .
The first partial derivatives are
∂f
= 6x2 y 2 ,
∂x
∂f
= 4x3 y + 3y 2 ,
∂y
and the second partial derivatives are
∂2 f
= 12xy 2 ,
∂x2
∂2 f
= 4x3 + 6y,
∂y 2
the last two being equal, as expected. ∂2 f
= 12x2 y,
∂x∂y
∂2 f
= 12x2 y,
∂y∂x
5.2 The total differential and total derivative
Having defined the (first) partial derivatives of a function f(x, y), which give the
rate of change of f along the positive x- and y-axes, we consider next the rate of
change of f(x, y) in an arbitrary direction. Suppose that we make simultaneous
small changes ∆x in x and ∆y in y and that, as a result, f changes to f + ∆f.
Then we must have
∆f = f(x + ∆x, y + ∆y) − f(x, y)
= f(x + ∆x, y + ∆y) − f(x, y + ∆y) + f(x, y + ∆y) − f(x, y)
f(x, y + ∆y) − f(x, y)
f(x + ∆x, y + ∆y) − f(x, y + ∆y)
∆x +
∆y.
=
∆x
∆y
(5.3)
In the last line we note that the quantities in brackets are very similar to those
involved in the definitions of partial derivatives (5.1), (5.2). For them to be strictly
equal to the partial derivatives, ∆x and ∆y would need to be infinitesimally small.
But even for finite (but not too large) ∆x and ∆y the approximate formula
∆f ≈
∂f(x, y)
∂f(x, y)
∆x +
∆y,
∂x
∂y
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(5.4)
PARTIAL DIFFERENTIATION
can be obtained. It will be noticed that the first bracket in (5.3) actually approximates to ∂f(x, y + ∆y)/∂x but that this has been replaced by ∂f(x, y)/∂x in (5.4).
This approximation clearly has the same degree of validity as that which replaces
the bracket by the partial derivative.
How valid an approximation (5.4) is to (5.3) depends not only on how small
∆x and ∆y are but also on the magnitudes of higher partial derivatives; this is
discussed further in section 5.7 in the context of Taylor series for functions of
more than one variable. Nevertheless, letting the small changes ∆x and ∆y in
(5.4) become infinitesimal, we can define the total differential df of the function
f(x, y), without any approximation, as
df =
∂f
∂f
dx +
dy.
∂x
∂y
(5.5)
Equation (5.5) can be extended to the case of a function of n variables,
f(x1 , x2 , . . . , xn );
df =
∂f
∂f
∂f
dx1 +
dx2 + · · · +
dxn .
∂x1
∂x2
∂xn
(5.6)
Find the total differential of the function f(x, y) = y exp(x + y).
Evaluating the first partial derivatives, we find
∂f
∂f
= y exp(x + y),
= exp(x + y) + y exp(x + y).
∂x
∂y
Applying (5.5), we then find that the total differential is given by
df = [y exp(x + y)]dx + [(1 + y) exp(x + y)]dy. In some situations, despite the fact that several variables xi , i = 1, 2, . . . , n,
appear to be involved, effectively only one of them is. This occurs if there are
subsidiary relationships constraining all the xi to have values dependent on the
value of one of them, say x1 . These relationships may be represented by equations
that are typically of the form
xi = xi (x1 ),
i = 2, 3, . . . , n.
(5.7)
In principle f can then be expressed as a function of x1 alone by substituting
from (5.7) for x2 , x3 , . . . , xn , and then the total derivative (or simply the derivative)
of f with respect to x1 is obtained by ordinary differentiation.
Alternatively, (5.6) can be used to give
∂f dx2
∂f dxn
df
∂f
=
+
+ ··· +
.
(5.8)
dx1
∂x1
∂x2 dx1
∂xn dx1
It should be noted that the LHS of this equation is the total derivative df/dx1 ,
whilst the partial derivative ∂f/∂x1 forms only a part of the RHS. In evaluating
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