An elementary treatise on the differential calculus, in which the method of limits is exclusively made use of, by the Rev. M. O'Brien.

60

CHANGE OF THE INDEPENDANT VARIABLE.

Change of easily do by remembering always, 1st, that the independant
pendant variable is that variable whose differential is constant; 2ndly,
variable; dy d'y W3y
how effect- 2 d3y
ed. dc t d x ' d~ &c. represent the successive derivatives
dxso du 2 d OB
of y considered as a function of v, on the express condition
that dx is a constant; and 3rdly, that these derivatives,
whatever be the independant variable, are represented by
dy
d (d
dy \d-x, dx
' dx dx - w e...&c &c.... respectively.
dxt do ldx
Hence, if we have any expression involving
dy d2y d3y
dx d'-v d3 '.."'
where these quantities are supposed to represent the successive
derivatives of y considered as a function of x, on which supposition x must necessarily be the independant variable: then
since
d d \(dx d x -
dx do dxc
represent the same derivatives, we may put these instead of
dy d~y d~y
dy d 2y d y3.. &c. respectively;
dx' d x" d x;'"
and then we may suppose any quantity we please to be the
independant variable, since the substituted expressions represent the successive derivatives of y, whatever be the independant variable. If the expression involve dx, d dy, dy, d3
&c. but not in the precise forms
dy d2y d'y
dx' dx2' dx' &c