Page 180 - Foundations Of Differential Calculus

8. On the Higher Differentiation of Differential Formulas 163
2
and for d y we everywhere write
2 2
2
dx d y − dx dy d x .
2
dx + dy 2
Hence, if the given expression is
2

dy dx + dy 2 ,
2
d x
2 2
in which dx + dy is assumed to be constant, then the expression is
transformed into
2 2 3/2
dx + dy
,
2
2
dy d x − dx d y
in which no differential is assumed to be constant.
276. In order that these reductions can be used more easily, we have
brought them together in the following table.
A differential formula of higher order can be transformed into one that
involves no constant differential by means of the following substitutions:

2
I. If the differential dx is assumed to be constant, then for d y we write
2
2 dy d x
d y −
dx
3
and for d y we write
2
2
3
2 2
3 3d xd y 3dy d x dy d x
d y − + 2 − .
dx dx dx
2
II. If the differential dy is assumed to be constant, then for d x we write
2
2 dx d y
d x − ,
dy
3
and for d x we write
2 2
3
2
2
3 3d xd y 3dx d y dx d y
d x − + − .
dy dy 2 dy
2
III. If the differential ydx is assumed to be constant, then for d x we write
dx dy
− ,
y