Page 159 - Foundations Of Differential Calculus

142 8. On the Higher Differentiation of Differential Formulas
Since dP = pdx and dp = qdx,wehave

3 3 2 3
d V = Pd x +3pdxd x + qdx .
We find the higher differentials in a similar way.
244. Now we apply this to powers of x, whose successive differentials we
investigate, supposing that dx is not kept constant.
I. If V = x, then
2 2 3 3 4 4
dV = dx, d V = d x, d V = d x, d V = d x, ....

2
II. If V = x , then
dV =2xdx
and
2 2 2
d V =2xd x +2dx ,
2
3
3
d V =2xd x +6dx d x,
4 4 3 2 2
d V =2xd x +8dx d x +6d x ,
5 5 4 2 3
d V =2xd x +10dx d x +20d xd x,
....

n
III. If in general V = x , then and
n−1
dV = nx dx,
2 n−1 2 n−2 2
d V = nx d x + n (n − 1) x dx ,
3
d V = nx n−1 3
d x
n−2 2 n−3 3
+3n (n − 1) x dx d x + n (n − 1) (n − 2) x dx ,
4 n−1 4 n−2 3 n−2 2 2
d V = nx d x +4n (n − 1) x dx d x +3n (n − 1) x d x
n−3 2 2
+6n (n − 1) (n − 2) x dx d x
n−4 4
+ n (n − 1) (n − 2) (n − 3) x dx ,
....

3
4
2
If dx happens to be constant, then d x =0, d x =0, d x = 0, and so
forth. Thus we have the same differentials we found earlier under this
hypothesis.