Page 132 - Foundations Of Differential Calculus

6. On the Differentiation of Transcendental Functions 115
VI. If y = e arcsin x , this formula can also be differentiated by the preceding
methods. Indeed, we have

dx
arcsin x
dy = e √ .
1 − x 2

In this way all functions of x involving not only logarithms and exponen-
tials, but also even circular arcs, can be differentiated.

200. Since the differentials of arcs when divided by dx are algebraic quan-
tities, their second, and higher, differentials can be found, as we have
shown, by differentiation of algebraic quantities. Let y = arcsin x. Since
√
2
dy = dx/ 1 − x ,wehave
dy 1
= √ ,
dx 1 − x 2
2
2
whose differential gives the value of d y/dx , provided that we keep dx
constant. Hence the differentials of this y of any order are of this kind.
If y = arcsin x, then

dy 1
= √ ,
dx 1 − x 2

and when we keep dx constant,

2
d y = x ,
dx 2 (1 − x ) 3/2
2
3
d y = 1+2x 2 ,
2
dx 3 (1 − x ) 5/2
4 3
d y = 9x +6x ,
dx 4 (1 − x ) 7/2
2
2
5
d y = 9+72x +24x 4 ,
2
dx 5 (1 + x ) 9/2
6 3 5
d y = 225x + 600x + 120x ,
dx 6 (1 − x ) 11/2
2
....