Page 167 - Foundations Of Differential Calculus

150 8. On the Higher Differentiation of Differential Formulas
2 2 2 2
I. Let y = x . Then dy =2xdx and d y =2xd x +2dx . When these
values are substituted into the formula
2
2
dy d x − dx d y ,
dx 3
we have
2 2 3
2xdxd x − 2xdxd x − 2dx = −2.
dx 3

n
II. Let y = x . Then dy = nx n−1 dx and
2 n−1 2 n−2 2
d y = nx d x + n (n − 1) x dx .
When these values are substituted, the formula
2
2
dy d x − dx d y
dx 3
is transformed into
n−1 2 n−1 2 n−2 3
nx dx d x − nx dx d x − n (n − 1) x dx = −n (n − 1) x n−2 .
dx 3
√
2
III. Let y = − 1 − x . Then
xdx
dy = √
1 − x 2
and 2 2
2 xd x dx
d y = √ + 3/2 ,
2
1 − x 2 (1 − x )
so that the formula
2
2
dy d x − dx d y
dx 3
becomes
2 2
xd x xd x 1 −1
√ − √ − 3/2 = 3/2 .
2
2
dx 2 1 − x 2 dx 2 1 − x 2 (1 − x ) (1 − x )
2
In all of these examples the second differentials d x cancel each other. This
also happens no matter what other functions of x are substituted for y.
258. Since these examples have already shown the truth of our proposi-
tion, namely, that the formula
2
2
dy d x − dx d y
dx 3