An elementary treatise on the differential calculus, in which the method of limits is exclusively made use of, by the Rev. M. O'Brien.

31i
41. If y3 + a2y-2 a3 + axy- x3 = then will
x x2 a 2 a2
y=a- - + -— &c. and y=x+ + - +-&c.
22 26a Y 9x
42. If y3 -ay + ay - 2X3 =, then will
a a2 a3 __ _ 2
y=x I -- + -+ - - + &c., y-=a+ - + -- + &c.
3 3x 81 *2 2 8a
<3 4 5 X6
and y = --- &c.
and a a5
43. It is required to prove that
1 1 1 1 1
log ( ) = - (=) + 3 ( &C.
44. Find the developement of u- = e6e to five terms.
45. Given the differential coefficients of it is
1 + 9
required to deduce the differential coefficients of log
1 + x
46. From the differential coefficients of cos x, it is required to derive those of log cos x.
47. Given the differential coefficients of log x, to deduce
those of log of log x, or of log2 X.
48. If u =f (), it is required to prove that
xdu x 2du (aI d3
f(2t) =f (X) + d + +.2.3 +&c.
vv2 dvl.^d 1.2.3 d^
xdu X2d2u Xa3d3
f (0) =f (X) - - + -2- - + &c.
Oda t 1.2 dx" 1.2.3 dxa
49. On the same hypothesis will there result
f =fsG) + du x+ d2u X"
f) (~) dt6 r d+ Jr&C.:
Z) ) d x 2 P 1.2.22
/a?\ du u d2u X2
f^ =f (.)d X'2 da 2 1 2.22