200 
DERIVATIVES AND INTEGRALS 
[VI 
and 
3. Show that 1 /(x 2 +1) = {\ij(x 4- ¿)} - [\ij(x - 0} 
l 1 /(1 + X s ) = {i/(^ + l)} + {s<o/(^ + a))} + {Jo) 2 /{x + w 2 )} 
where w and w 2 are the complex cube roots of unity. Write down the 
derivatives of the functions and verify that they agree with the results 
obtained by means of the formula R'=(P'Q— PQ')/Q 2 , 
4. If Q has a factor (x — a) m , the denominator of R' (when reduced to its 
lowest terms) is divisible by (x — a) m +1 but by no higher power of (x — a). 
5. In no case can the denominator of A" have a simple factor x -a. 
Hence no rational function (such as l/x) whose denominator contains any 
simple factor can be the derivative of another rational function. 
99. C. Algebraical Functions. We add a few examples 
connected with the differentiation of irrational algebraical func 
tions. At present we are not in a position to consider this 
question systematically, as the theorems which we have proved do 
not furnish ns with any simple or convenient method of finding 
the derivative even of so elementary a function as V(1 +# 2 ). The 
further consideration of the subject must be postponed until 
108-9. 
Examples XLV. 1. The derivative of x m is rnx m ~ l for any rational 
value of m (§ 98). Deduce from Th. (8) of § 94 that the derivative of 
(ax + b) m is ma(ax+b) m ~ 1 . 
2. Find the derivatives of 
x /(!+^), v/(l—#), y/{{l+x)/(l-#)}, K l{(ax+b)/(cx + d)}, {(1 +x)/(l ~x)} m , 
(1 + sjx)!{ 1 - Jx), {J(x +1)+J(x -1 )}!{J{x +1) - J{x - 1)}, (ax + b) m (cx + d) n . 
3. Show that if y = Jx+>J(x+ t Jx) then ^=y 4 /(2?/ + l) 2 . Show that 
dxjdy = 4?/ 3 (y-f l)/(2y + l) 3 and deduce that 
dy _ 1 
dx 2Jx 
100. D. Transcendental Functions. We have already 
proved (Ex. xli. 5) that 
D x sin x = cos x, D x cos x = — sin x. 
By means of Theorems (4) and (5) of § 94 the reader will 
easily verify the following formulae: 
D x tan x — sec 2 x, D x cot x = — cosec 2 x, 
D x sec x = tan x sec x, D x cosec x = — cot x cosec x. 
By means of Theorem (8) the reader can write down the 
derivatives of tan {ax + b), cot (ax + b), etc. And by means of 
Theorem (9) we cari determine the derivatives of the ordinary