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

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5. If u =(a + x) -, then will
1 4x o102 20x 35 x4
- = + -'+ -- &c.
a4 a a6 a' a8
6. If u = (a + x)~, then will u=-a multiplied into
1 c 1.2 x2 1.2.5 x3 1.2.5.8 t4
+23 --- + - C.
3a 3"2 lo2a2 33" 1.2.3 a - 3" 1.2.3.44a J
axc a2X 2 a3 3
7.f it = e-a, then u =1 - + -- --- + &c.
71 12- 1.2.3
8 If u = log (a + bx), then will
b x b22 b3 x3 b4 x4
u = log a + — - &
alo~ a1 2 a 3 - 4
9. If log -ta b then will
\a-boGJ
(b b3 3 b' x b5 7 t7
u=2 + 5, + &c.
\a 43 3 a 5 c 7 /
10. If u = sin (a + bx), then will
2 3
usina+bc 1 1.2 2.3
1 102 1.2.3
11. If u -cos (ax -b), then will
cos b -a sin b - - cos b --- -a3 sin b - - &c.
1 1.2 1.2.3
12. If u==sin- -, then will: — - -+ + &c.
a a 1.2 a
x 2 X3 24 5
13. If u =tan, then u= - + - + - + &c:
1 1.2.3 1.2.3.4.5
14. If =tan- -, then will
a
00: IV<?3 C5 G7
u5. -- --- + -- - &c.
l.a 3. a3 5. a3 7. a1
15. If u=esi, then u + - &C —
1 1.2 2.4