the calculus of functions « 
Particular cases of 4 4 oc = x 
235 
I X 
l\)X = 
2 — x 
AT 1 
X 
4*27 = 
I -j-^ 
I X 
4a? 
= (— r 
\ 2 — x n f 
1 
2 a % 
(2 at” 2) n 
4a? — 
2 ac — c^x 
XJtAJ 
X 
a+bx 
~¥Tc % r 
C x 
4<2?= 
:log 2 — rr-J-1' 
2 a 
of 4 6 X : 
= X 
fyx — 
4# = 3 
3( 
1 », r _ 3fZZ 
1 — *) “ 3 X 
3—x 
x~ l 
X 
4a? 
4a? : 
3 fl 
3 ac — c x 
3 + 3* 
4a? 
4 / x 
a + bx 
b 2, — bc+c 7 
c — x 
3* 
X n — 
x \ 3 
1 
I \ » 
3— AT 
yj/X =log3 x + log ( £ * *) 
Problem XXXIII. 
Given the equation 
4 '# x — 
d \ ]/ x 
dx 
a being such a function that a* x = x 
For x put ax, then 
,0 , d 4" «• i 
4/a x =3 4 X = — - 
T T a « X 
by differentiating this we have 
d \ \ x d. d a x 
O) 
tLr dx dax 
but the left side of this equation is by the Problem equal to 
ypax; therefore 
but we also have 
consequently 
. d. d 4* a X 
' dx dax 
d 4/ a x d -i/ a X da, X 
dx 
d a x dx 
d 4^ a x d + ax Id ax 
dxx dx \ dx 
I i 
MDCCCXVI. 
