98] DERIVATIVES AND INTEGRALS 199 
the sum of a number of terms of the type 
A /(x — a) p 
where a is a root of Q(x) = 0. We know already how T to find the 
derivative of the polynomial ; all that remains, therefore, is that 
we should determine the derivative of the function written above, 
in which, it must be remembered, a may be real or complex. 
We shall first establish the following proposition : the de 
rivative of x m is mx m_1 , for all integral or rational values 
of m, positive or negative. 
We have already seen (Ex. xli. 4) that this is so when m is 
a positive integer. The more general result (which was proved 
indirectly in Ex. xlii. 5) is a mere corollary from Ex. xxxvn. 4, 
For, if (f) (x) — x m , 
<p'(x) = lim \{x + h) m — x m ]/h 
= lim (£ m — x m )/{£ — x) = mx m ~ l . 
From this result it follows at once, by Theorem (6) (§ 94) that if 
(f) (x) = A/(x — a) p then (f> r (x) = — pAj(x — a)^ +1 , provided a is real. 
We shall however require the result for complex as well as for real values 
of a. We shall therefore give a direct proof applicable to the case in which 
p is a positive integer and a has any value whatever. Then 
{(# + h - a) ~ p -{x- a)~ p \lh — -{x + h — a)~ p {x -a)~ p {{x + h — a) p — (a? — a) p )jk 
which, when we expand (x + h-a) p in powers of h by the Binomial Theorem, 
and make h tend to 0, obviously tends to 
- {x - a)~ 2p [p{x - a) p ~ l } = -p{x-a)~ p ~ 1 . 
We are now able to write down the derivative of the general 
rational function R{x), in the form 
TJY \ -4i,i 2^4] >2 An t i 2Ao t 2 
' ' (x — «i) 2 (x — Qj) 3 (#-ot 2 ) 2 (x-a 2 ) 3 
Examples XLIV. 1. Differentiate 
xKl-x 2 ), (l+^lil-x 2 ), {(1 +x)l{l-x)Y, (l-x+x*)l{l+x+x*), 
x{l-x)l(l+x 3 ), ^(l+a- 2 )/(I+.v 4 ), a’(l-.r 2 )/(l+^ 2 + .r 4 ), 
Il{x — a){x—b){x—c), {x- a)/{x-b)(x—c), {x-a){x — b)j{x — c)(x — d), 
x/{x 2 + a 2 ){x 2 + b 2 ), {x 2 + a 2 )/{x 2 + b 2 ), {x- a) 2 /(x+a) 2 {x 2 + b 2 ). 
2. The derivative of (ax 2 + 2bx + c)/{Ax 2 + 2Bx + C) is 
{{ax + b){Bx + 0- {bx+c){Ax + B)}/{Ax 2 + 2Bx+C) 2 .