Page 106 - Foundations Of Differential Calculus

5. On the Differentiation of Algebraic Functions of One Variable 89
Frequently, it may be more expeditious to use the rule in its earlier form

p dp pdq
d. = − ,
q q q 2
so that the differential of a quotient is equal to the quotient of the dif-
ferential of the numerator by the denominator minus the quotient of the
product of the differential of the denominator and the numerator by the
square of the denominator. From this we have:
2
a − x 2
III. If y = , then
2 2
4
a + a x + x 4
2 2 2 3
−2xdx a − x 2a xdx +4x dx
dy = − 2 ,
2 2
4
2 2
4
4
a + a x + x 4 (a + a x + x )
and when we take a common denominator we have
4 2 2 4
−2xdx 2a +2a x − x
dy = 2 .
4
4
2 2
(a + a x + x )
165. This should be sufficient for the investigation of differentials of ra-
tional functions. If the function happens to be a polynomial, we have said
enough. If the function is a quotient, it can always be reduced to the fol-
lowing form:
2 3 4 5
A + Bx + Cx + Dx + Ex + Fx + ···
y = .
5
3
4
2
α + βx + γx + δx + x + ζx + ···
We let the numerator be equal to p and the denominator be equal to q,so
that y = p/q and
qdp − pdq
dy = .
q 2
But since
2 3 4
p = A + Bx + Cx + Dx + Ex + ···
and
2 3 4
q = α + βx + γx + δx + x + ··· ,
we have
2 3
dp = Bdx +2Cx dx +3Dx dx +4Ex dx + ···
and
2 3
dq = βdx +2γx dx +3δx dx +4 x dx + ··· .