Page 115 - Foundations Of Differential Calculus

98 5. On the Differentiation of Algebraic Functions of One Variable
1
II. Let y = √ ; find the first- and higher-order differentials.
1 − x 2
dy x
= ,
2
dx (1 − x ) 3/2
2 2
d y = 1+2x ,
2
dx 2 (1 − x ) 5/2
3
d y 9x +6x 3
2
dx 3 = (1 − x ) 7/2 ,
4 2 4
d 9+72x +24x
dx 4 = (1 − x ) 9/2 ,
2
3
5
d y 225x + 600x + 120x 5
2
dx 5 = (1 − x ) 11/2 ,
6
2
4
d y 225 + 4050x + 5400x + 720x 6
2
dx 6 = (1 − x ) 13/2 ,
and so forth. These differentials can easily be continued, but the law
by which the terms proceed may not be immediately obvious. The
coefficient of the highest power of x is the product of the natural num-
bers from 1 to the order of the differential. Meanwhile, if we wish to
√
2
continue further our investigation, we will find that if y =1/ 1 − x ,
generally we have
n
d y 1 · 2 · 3 ··· n
= 1
2
dx n (1 − x ) n+ 2

1 n (n − 1) n−2
n
× x + · x
2 1 · 2
1 · 3 n (n − 1) (n − 2) (n − 3) n−4
+ · x
2 · 4 1 · 2 · 3 · 4
1 · 3 · 5 n (n − 1) ··· (n − 5) n−6
+ · x
2 · 4 · 6 1 · 2 ··· 6
1 · 3 · 5 · 7 n (n − 1) ··· (n − 7) n−8
+ · x + ··· .
2 · 4 · 6 · 8 1 · 2 ··· 8

Examples of this kind are useful not only for acquiring a habit of differ-
entiating, but they also provide rules that are observed in differentials of
all orders, which are very much worth noticing and can lead to further
discoveries.