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

58,

SUCCESSIVE DIFFERENTIATION.

ddd... (to n d's)y
f X)= dxn
To avoid the repetition of d's we represent d repeated n
times by d", and thus we have
f " (X) it
d Xy
For examples of finding successive differential coefficients,
see Appendix (D).
Expression 98. We have supposed dx constant for the sake of simfor f- (x)
when d x plicity; if we do not make this supposition, then we have
is not supposed con- dI
stant. ____
( dy d dd
dx] df'(x) ________.. &c.
J X)= - f _ _____=__
d dxxd dx
which are the expressions for the successive derivatives of
y considered as a function of v, whatever be the values of
dx and dy. If we perform the differentiations, and put d' for
dd, di for ddd, &c., &c., we have
{dy\ dydX - dy d'x d'y dv - dy d 2X
dx) dx',amd...f 2(& d
and.f~(x) =
df?) (d y dx - dy d'x) d' - (d'yd - dyd'x)3d 'dix
(d y dx - dy d v) dx - 3 (d'y dx - dy d ') d'x
d x5
and so we may go on. The expressions however become extremely complicated.
In these expressions, if we suppose dv constant, we have
d2X = 0, d'v = 0, &c.... and therefore
d'y d3y
f 2 ( =dx2 f3 () l "'