Page 110 - Foundations Of Differential Calculus

5. On the Differentiation of Algebraic Functions of One Variable 93
n
If, on the other hand, y = p/q , then by the general rule,
n
q dp − npq n−1 dq
dy = .
q 2n
If we divide both numerator and denominator by q n−1 we have
qdp − np dq
dy = .
q n+1
We conclude:
p
II. If y = , then
q n
qdp − np dq
dy = .
q n+1

m
n
If the given quotient is y = p /q , then we find that
mp m−1 n m n−1 dq
q dp − np q
dy = ,
q 2n
which reduces to
m−1 m
mp qdp − np dq
dy = .
q n+1
It follows that:
p m
III. If y = , then
q n
m−1
p (mq dp − np dq)
dy = .
q n+1
m n
Finally, if the given quotient is y = r/(p q ), then by the general
quotient rule we have
m n
p q dr − mp m−1 n m n−1 rdq
q rdp − np q
dy = .
q
p 2m 2n
m−1 n−1
Since both numerator and denominator are divisible by p q :
r
IV. If y = , then
m n
p q
pq dr − mqr dp − npr dq
dy = .
p m+1 n+1
q