Page 162 - Foundations Of Differential Calculus

8. On the Higher Differentiation of Differential Formulas 145
Let V = xy, so that

dV = ydx + xdy,
2 2 2
d V = yd x +2dx dy + xd y,
3 3 2 2 3
d V = yd x +3dy d x +3d ydx + xd y,
4 4 3 2 2 3 4
d V = yd x +4dy d x +6d xd y +4dx d y + xd y,
....

In this example the numerical coefficients follow the law of the powers of a
binomial, so that this can be continued howsoever far one wishes.
If V = y/x, then
dy ydx
dV = − ,
x x 2
2 2 2
2 d y 2dx dy 2ydx yd x
d V = − + − ,
x x 2 x 3 x 2
3 2 2 2
3 d y 3dx d y 6dx dy 3dy d x
d V = − + −
x x 2 x 3 x 2
3
2
6ydx d x 6ydx 3 yd x
+ − − ,
x 3 x 4 x 2
....
In this example the sequence of differentials is not as clear as in the previous
example.
250. This method of differentiation is not confined only to finite functions.
It can also be extended to any expression that already contains differentials.
The differential can be found whether or not some differential is assumed to
remain constant. Since each differential is differentiated by the same laws
as finite quantities, the rules given in the preceding chapters are still valid
and should be observed. Let V denote such an expression that we need to
differentiate, whether it is finite or infinitely large or infinitely small. The
method of differentiation can be seen from the following examples.
2
2

I. Let V = dx + dy . Then
2 2
dx d x + dy d y
dV = .
2
dx + dy 2
ydx
II. Let V = . Then
dy
2
2
yd x ydx d y
dV = dx + − .
dy dy 2