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

190

th
II DIFFERENTIAL COEFF~ICIENT

This is called Leibnitz' Theorem, and is often of use in
finding successive differential coefficients.
nth differ- 20 oW
ential co- 260 Tofind the nth diJerentiat coeficient of f (x) if
efacienrt of f (x) = q (a + b x + cx2).
cp (a-tbx
+ cx').. We have f v + ~h) = (A+ Bh + ch) (...(),
if A = a + bx.+ C, X2 B =b~+ 2cw.
Now the general term of / (A + Bh + ch'2) expanded in
powers of Bh + Ch2 is
(Bh + ch"r
Fr
and the general term of this expanded in powers of chM is
Bil - SCS
(Pr(a) s.F(r -s) +s
In this term let us suppose that r + s = n, and it becomes
B' r - a 7- r
orr(A) Bh-e-r h-= U,.h", suppose.
r/r()1 (n - r) 17 (2 r - n)A'=Uhspoe
Now s is evidently not > r, and not <0, i. e. n - r is not
n
> r and not < 0, i. e. r is not < - and not > n.
Hence the sum of the terms formed by giving r all possible values is
(Un + Un - + U,... -.2 U.) h' if n be even;
2
and (U,, + U, -1+ U,,... -2 U,) h if n be odd;
and hence it follows that the coefficient of h" in the second
member of (1) expanded in powers of h is
U, + U,,-, + Ufl,2... &C. o or -
2 '2