Page 158 - Foundations Of Differential Calculus

8

On the Higher Differentiation
of Differential Formulas

242. If there is a single variable and its differential is held constant, we
have already given the method for finding differentials of any order. That
is, if the differential of any function is differentiated again, we obtain its sec-
ond differential. If this is again differentiated, we get the third differential,
and so forth. This same rule holds whether the function involves several
variables or only one, whose first differential is not kept constant. Hence,
if V is any function of x and dx is not held constant, but is as if it were
2
a variable, then the differential of dx is equal to d x. The differential of
2
3
d x is equal to d x, and so forth. Let us investigate the second and higher
differentials of the function V .
243. We let the first differential of the function V be equal to Pdx,
where P is some function of x depending on V .Ifwewanttofindthe
second differential of V , we must differentiate again its first differential
Pdx. Since this is the product of two variable quantities, P and dx, whose
2
differentials respectively are dP = pdx and d.dx = d x, then according to
the product rule the second differential is

2
2
2
d V = Pd x + pdx .
2
2
Then if we let dp = qdx, since the differential of dx is equal to 2dx d x,
we have by another differentiation
3 3 2 2 2
d V = Pd x + dP d x +2pdxd x + dp dx .