Page 96 - Foundations Of Differential Calculus

5. On the Differentiation of Algebraic Functions of One Variable 79
and generally,
1 −m
= x .
x m
If we substitute in the previous formula −m for n, we have the first differ-
ential of 1/x m equal to
−mdx ;
x m+1
the second differential is equal to

m (m +1) dx 2 ;
x m+2
the third differential is equal to

−m (m +1) (m +2) dx 3 ;
x m+3
and so forth. The following simpler cases deserve to be noted:

1 −dx 2 1 2dx 2 3 1 −6dx 3
d. = , d . = , d . = , ...,
x x 2 x x 3 x x 4
1 −2dx 2 1 6dx 2 3 1 −24dx 3
d. = , d . = , d . = , ...,
x 2 x 3 x 2 x 4 x 2 x 5
2 3
1 −3dx 2 1 12dx 3 1 −60dx
d. = , d . = , d . = , ...,
x 3 x 4 x 3 x 5 x 3 x 6
1 −4dx 2 1 20dx 2 3 1 −120dx 3
d. = , d . = , d . = , ...,
x 4 x 5 x 4 x 6 x 4 x 7
1 −5dx 2 1 30dx 2 3 1 −210dx 3
d. = , d . = , d . = , ...,
x 5 x 6 x 5 x 7 x 5 x 8
and so forth.

156. Then if we let n be a fraction, we obtain differentials of irrational
√
expressions. If n = µ/ , then the first differential of x µ/Ð , that is ν x ,is
µ
equal to
µ (µ−ν)/ν µ √
ν
x dx = dx x µ−ν .
ν ν
The second differential is equal to

µ (µ − ν) x (µ−2ν)/ν dx = µ (µ − ν) dx 2 ν x µ−2ν ,
√
2
ν 2 ν 2