Page 127 - Foundations Of Differential Calculus

110 6. On the Differentiation of Transcendental Functions
x
p
e
x
Then dy = e dp, and since dp = e dx, it follows that if y = e , then
x
x
e
dy = e e dx.
If y = e e e x , then
x
x
e
dy = e e e x e e dx.
r
q
z
r
However, if y = p , then we let q = z, and dy = p dz ln p + zp z−1 dp, but
r
dz = q dr ln q + rq r−1 dq, so that
z r
z
z r
dy = p q dr ln p · ln q + p rq r−1 dq ln p + p q dp .
p
r
q
It follows that if y = p , then
rdq ln p dp
r
q
dy = p q r dr ln p · ln q + + .
q p
In this way, no matter how the exponential may occur, the differential can
be found.
194. We proceed now to transcendental quantities. Previously, a consid-
eration of circular arcs has led us to a knowledge of these. Let an arc of
a circle whose radius is always equal to unity be given, and let the sine of
this arc be equal to x. We express this arc as arcsin x and we investigate
the differential of this arc, that is, the increment that it receives if the sine
of x is increased by its differential dx. We can accomplish this by the differ-
entiation of logarithms, since in Introduction (loc. cit., paragraph 138) we
have shown that the expression arcsin x can be reduced to this logarithmic
expression:
1 √
2
√ ln 1 − x + x −1 .
−1
We let y = arcsin x, so that
1 √
2
y = √ ln 1 − x + x −1 ,
−1
whose differential we have seen (paragraph 182, VII) to be
√
1 −xdx √ √
√ √ + dx −1 2
−1 1−x 2 dx x −1+ 1 − x
dy = √ √ = √ √ √ ,
2
2
1 − x + x −1 1 − x + x −1 1 − x 2
so that
dx
dy = √ .
1 − x 2