Page 109 - Foundations Of Differential Calculus

92 5. On the Differentiation of Algebraic Functions of One Variable
especially since we will soon give a general method that will comprehend
all of these special methods of differentiating.
167. There are some cases in which the differential can be more easily
expressed than with the general rules we have given; these would be either
products or quotients in which the factors that make up the product or the
numerator or denominator of the quotient are powers.
m n
We suppose that the function that is to be differentiated is y = p q .
n
To find the differential of this function we let p m = P and q = Q, so that
y = PQ and dy = PdQ + QdP.

Since
m−1 n−1
dP = mp dp and dQ = nq dq,
when we substitute these values, we obtain
m n−1 m−1 n m−1 n−1
dy = np q dq + mp q dp = p q (np dq + mq dp) .
From this result we derive the following rule:

m n
I. If y = p q , then
m−1 n−1
dy = p q (np dq + mq dp) .
In a similar way, if there are three factors, the differential can be found
and expressed as follows:
m n k
II. If y = p q r , then
m−1 n−1 k−1
dy = p q r (mqr dp + npr dq + kpq dr) .

168. If a quotient has either a numerator or a denominator that has a
factor that is a power, we can give special rules.
m
First we suppose that the quotient has the form y = p /q. Then from
the general rule for quotients we have
m
mp m−1 qdp − p dq
dy = ,
q 2
but this differential can be expressed more conveniently as:

p m
I. If y = , then
q
p m−1 (mq dp − pdq)
dy = .
q 2