Page 119 - Foundations Of Differential Calculus

102 6. On the Differentiation of Transcendental Functions
1 √ √ 2 √
VII. If y = √ ln x −1+ 1 − x ,welet z = x −1. Also, since y =
−1
1 √ 2
√ ln z + 1+ z , by the previous example we have
−1
1 dz
dy = √ √ .
−1 1+ z 2
√
Since dz = dx −1, we have
dx
dy = √ .
1 − x 2
Although the given logarithm involves a complex number, the differ-
ential is real.

183. If the logarithm of a product is given, then the logarithm is expressed
as a sum in the following manner. If y =ln pqrs is given, since y =ln p +
ln q +ln r +ln s,wehave
dp dq dr ds
dy = + + + .
p q r s
This reduction also has a use if the logarithm of a quotient is to be differ-
entiated. If
pq
y =ln ,
rs
since y =ln p +ln q − ln r − ln s,wehave

dp dq dr ds
dy = + − − .
p q r s
Powers give no more difficulty. If
m n
p q
y =ln ,
µ ν
r s
since y = m ln p + n ln q − µ ln r − ν ln s,wehave
mdp ndq µdr νds
dy = + − − .
p q r s
I. If y =ln (a + x)(b + x)(c + x), since

y =ln (a + x) + ln (b + x)+ln (c + x) ,

the desired differential is
dx dx dx
dy = + + .
a + x b + x c + x