Page 108 - Foundations Of Differential Calculus

5. On the Differentiation of Algebraic Functions of One Variable 91
pr
I. If y = , then
q
pq dr + qr dp − pr dq
dy = .
q 2
p
If y = ,welet qs = Q, so that
qs
dQ = qds + sdq
and

Qdp − pdQ
dy = .
2 2
q s
It follows that:
p
II. If y = , then
qs
qs dp − pq ds − ps dq
dy = .
2 2
q s
pr P
If y = , again, we let pr = P and qs = Q, so that y = and
qs Q
QdP − PdQ
dy = .
Q 2

Since
dP = pdr + rdp and dQ = qds + sdq,

we obtain the following differentiation:
pr
III. If y = , then
qs
pqs dr + qrs dp − pqr ds − prs dq
dy = ,
2 2
q s
or
rdp pdr pr dq pr ds
dy = + − − .
2
qs qs q s qs 2
In a similar way, if the numerator and denominator of the quotient con-
tained several factors, using the same reasoning we could investigate the
differential. It does not seem to be necessary that one be led by hand
through the argument. For this reason we omit any examples of this kind,