Page 121 - Foundations Of Differential Calculus

104 6. On the Differentiation of Transcendental Functions
It follows that

2 2
dp dq −q pdx − p + q dx
− = √ − √ = √ .
p q 2p 1 − x 2 2q 1 − x 2 2pq 1 − x 2
2 2
Since p + q = 4 and pq =2x,wehave
dx
dy = − √ .
x 1 − x 2
This differential can more easily be found if the given logarithm is trans-
formed by rationalization as follows:
√
1+ 1 − x 2 1 1
y =ln =ln + − 1 .
x x x 2
If we let

1 1
p = + − 1,
x x 2
then
√

−dx dx −dx dx −dx 1+ 1 − x 2
dp = − = − √ = √ .
x 2 3 1 x 2 x 2 1 − x 2 x 2 1 − x 2
x x 2 − 1
Since √
1+ 1 − x 2
p = ,
x
we have
dp −dx
dy = = √ ,
p x 1 − x 2
as we have already seen.

184. Since the first differentials of logarithms, when divided by dx, are
algebraic quantities, the second differentials and those of higher orders can
easily be found with the rules of the previous chapter, provided that we
assume that the differential dx is constant. Hence, if we let y =ln x, then
dx dy 1
dy = and = ,
x dx x
2
2 −dx 2 d y −1
d y = and = ,
x 2 dx 2 x 2
3 3
3 2dx d y 2
d y = and = ,
x 3 dx 3 x 3
4
4 −6dx 4 d y −6
d y = and = ,
x 4 dx 4 x 4