Page 113 - Foundations Of Differential Calculus

96 5. On the Differentiation of Algebraic Functions of One Variable
as we have seen before. In a similar way if the given function is
pq
y = ,
rs
we let each of the parts successively be variable and obtain the following
differentials:
qdp pdq −pq dr −pq ds
, , , .
2
rs rs r s rs 2
It follows that
qrs dp + prs dq − pqs dr − pqr ds
dy = .
2 2
r s
174. Provided only that each of the parts that make up a function are
such that we can find their differentials, we can find the differential of the
whole function. Hence, if the parts are integral powers, we can find their
differentials not only by means of the laws we have given before, but also
from this general rule. If the parts are irrational, since the irrationality
comes from the fractional exponents, we can differentiate these through
n
the differentiation of powers, that is, d.x = nx n−1 dx. From this same well
we draw the differentiation of like irrational formulas that involve other
surds. Therefore, it should be clear that if with this general rule, which
will be proved later, we join the rule for differentiating powers, then the
differentials of absolutely all algebraic functions can be exhibited.
175. From all of this it clearly follows that if y is any [algebraic] function
of x, its differential dy will have the form dy = pdx, where p can always
be found from the laws we have set down. Furthermore, the function p
is also an algebraic function of x, since in determining the differential no
other operations were used except the usual ones for algebraic functions.
For this reason if y is an algebraic function of x, then dy/dx is also an
algebraic function of x. Furthermore, if z is an algebraic function of x, such
that dz = qdx, since q is an algebraic function of x, we also know that
dz/dx is an algebraic function of x, and indeed so is dz/dy an algebraic
function of x which is equal to p/q. Hence, if the formula dz/dy is part of
some algebraic expression, this does not prevent the whole expression from
being algebraic, provided only that y and z are algebraic functions.
176. We can extend this line of reasoning to second- and higher-order
differentials. If y is an algebraic function of x, dy = pdx and dp = qdx,
2
2
then with dx remaining constant, we have d y = qdx , as we have already
seen. Since for the reasons already given q is also an algebraic function of
2
2
x, it follows that d y/dx is not only a finite quantity but also an algebraic
function of x, provided only that y is such a function. In a similar way we
see that
3
4
d y , d y , ...
dx 3 dx 4