Page 160 - Foundations Of Differential Calculus

8. On the Higher Differentiation of Differential Formulas 143
245. Since differentials of any order are differentiated according to the
same rules as are finite quantities, any expression in which differentials
occur besides finite quantities can be differentiated according to the rules
given above. Since this operation occurs not infrequently, we illustrate this
with a few examples.

2
xd x
I. If V = , we differentiate to obtain
dx 2
3
2
2 2
xd x d x 2xd x
dV = + − .
dx 2 dx dx 3
x
II. If V = , then
dx
2
xd x
dV =1 − 2 ,
dx
where there is no problem if we let V be an infinite quantity.

III. If
2
2 d x
V = x ln ,
dx 2
2 2 2
we first transform V into x ln d x−2x ln dx, and then by the ordinary
rules for differentiating we have
2 3 2 2
2 x d x 2x d x
dV =2xdx ln d x + − 4xdx ln dx − .
2
d x dx
The higher differentials of V can be found in a similar way.
246. If the given expression involves two variables, namely x and y, either
only one of the differentials can be kept constant, or neither. It is arbitrary
which of the differentials is assumed to be constant, since it depends on
our choice of the extent to which we want successive values to increase.
However, we cannot decide to keep both differentials constant, since this
would assume some relationship between x and y, while we have assumed
that there is no such relationship, or if there is, that it is unknown. If
we suppose that x increases equally and y also takes equal increments,
then by that fact we would have y = ax + b, and hence y would depend
on x, which is contrary to the hypothesis. For this reason either only one
differential of a variable is kept constant, or neither is kept constant. If we
know how to perform differentiations with no differential taken as constant,
it is clear how to find differentials if one differential is kept constant: If dx
2 3 4
is constant, we need only let the terms that contain d x, d x, d x, and so
forth, be deleted.