Page 122 - Foundations Of Differential Calculus

6. On the Differentiation of Transcendental Functions 105
etc. If p is an algebraic quantity and y =ln p, then although y is not an
algebraic quantity, nevertheless,
3
2
dy d y d y
, , ,
dx dx 2 dx 3
etc., are algebraic functions of x.
185. Now that we have discussed the differentiation of logarithms, those
functions that are a combination of logarithms and algebraic functions are
easily differentiated. Those functions that consist only of logarithms can
also be differentiated, as is clear from the following examples.
2 2
I. If y = (ln x) ,welet p =ln x, and since y = p ,wehave dy =2pdp.
but dp = dx/x, so that
2dx
dy = ln x.
x

n
II. In a similar way, if y = (ln x) , then
ndx n−1
dy = (ln x) ,
x
√
1
so that if y = ln x, since n = ,wehave
2
dx
dy = √ .
2x ln x
n
III. If p is any function of x and y = (ln p) , then
ndp n−1
dy = (ln p) .
p
Hence, since the differential dp can be found by our previous work,
the differential of y itself is known.

IV. If y = (ln p) (ln q) with p and q being any functions of x,bythe
product rule given before,

dp dq
dy = ln q + ln p.
p q

V. If y = x ln x, then by the same rule,
xdx
dy = dx ln x + = dx ln x + dx.
x