Page 126 - Foundations Of Differential Calculus

6. On the Differentiation of Transcendental Functions 109
as we have shown before in the rule given in paragraph 187. When we
q
replace e q ln p with p ,wehave
q
p qdp q q−1
q
dy = p dq ln p + = p dq ln p + qp dp.
p
x
x
If y = x ,wehave dy = x dx ln x. From this its higher differentials can
be defined. We see that
2
d y x 1 2
dx 2 = x x +(1+ln x) ,
3
d y x 3 3 (1+ln x) 1
dx 3 = x (1+ln x) + x − x 2 ,
etc.

192. Among the differentials of this kind of function, which involve expo-
nential functions, the following examples should be especially noted. They
x
arise from the differentiation of e p, indeed,
x
x
x
x
d.e p = e dp + e pdx = e (dp + pdx) .
x n
I. If y = e x , then
x n
x
dy = e nx n−1 dx + e x dx,
or,
x

dy = e dx nx n−1 + x n .
x
II. If y = e (x − 1), then
x
dy = e x dx.
2

III. If y = e x x − 2x +2 , then
x 2
dy = e x dx.
3
2

IV. If y = e x x − 3x +6x − 6 , then
x 3
dy = e x dx.
193. If the exponents themselves are again exponential quantities, then
differentiation is accomplished according to the same rules. Thus, if we
x
e
x
want to differentiate e ,welet p = e , so that
p
y = e e x = e ;