Page 130 - Foundations Of Differential Calculus

6. On the Differentiation of Transcendental Functions 113
199. Although the arc whose sine or cosine or tangent or cotangent or
secant or cosecant or finally versed sine is given is a transcendental quantity,
nevertheless its differential when divided by dx is an algebraic quantity. It
follows that its second differential, its third, fourth, and so forth, when
divided by the appropriate power of dx, are also algebraic. In order that
this differentiation might better be seen, we adjoin some examples.
√ √
2
2
I. If y = arcsin 2x 1 − x ,welet p =2x 1 − x , so that y = arcsin p
2

and dy = dp/ 1 − p . But then
2

2
2x dx 2dx 1 − 2x 2
dp =2dx 1 − x − √ = √
1 − x 2 1 − x 2
2
2
and 1 − p =1 − 2x , so when these values are substituted we have
2dx
dy = √ .
1 − x 2
√
2
From this it is clear that 2x 1 − x is the sine of twice the arc where
x is the sine of the original arc. Hence if y = 2 arcsin x, then dy =
√
2
2dx/ 1 − x .
II. If
1 − x 2
y = arcsin ,
1+ x 2
we let
1 − x 2
p = ,
1+ x 2
so that
−4xdx 2x
2
dp = 2 and 1 − p = 2 .
2
(1 + x ) 1+ x
Since
dp
,
dy = 2
1 − p
we have
−2dx
dy = .
1+ x 2

III. If y = arcsin (1 − x)/2, we let p = (1 − x)/2, so that

2 1+ x −dx
1 − p = and dp = .
2 1−x
4
2
It follows that
dp −dx
= √ .
dy = 2 2
1 − p 2 1 − x