Page 118 - Foundations Of Differential Calculus

6. On the Differentiation of Transcendental Functions 101
182. By means of this rule the differential of the logarithm of any given
function of x whatsoever is easily found, which will be clear from the fol-
lowing examples.
I. If y =ln x, then
dx
dy = .
x
n
n
II. If y =ln x ,welet x = p, so that y =ln p and dy = dp/p. But
dp = nx n−1 dx, so that
ndx
dy = .
x
The same result can be found from the nature of logarithms; since
n
ln x = n ln x,wehave
ndx
n
d ln x = nd ln x = .
x
2
III. If y =ln 1+ x , then
2xdx
dy = .
1+ x 2
1
IV. If y =ln √ , since
1 − x 2
1 2
2
y = − ln 1 − x = − ln 1 − x ,
2
we see that
xdx
dy = .
1 − x 2
x 1 2
V. If y =ln √ , since y =ln x − 2 ln 1+ x ,wehave
1+ x 2
dx xdx dx
dy = − = .
2
x 1+ x 2 x (1 + x )
√
2
VI. If y =ln x + 1+ x ,wehave

√ √
dx + xdx 1+ x 2 xdx + dx 1+ x 2
dy = √ = √ √ ;
x + 1+ x 2 x + 1+ x 2 1+ x 2
but since both numerator and denominator of this fraction are divis-
√
2
ible by x + 1+ x ,wehave
dx
dy = √ .
1+ x 2