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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56. It is required to prove by Taylor's Theorem, that
h h2
tan (x + h) = tan x + sec2xR - + 2 sin x sec3x - + &c.
1 1.2
57. Shew by the same Theorem, that
1 h x h2
sin-' (x + h) = sin- x (- 2- -+ --
sin'(+/,)=sin'+ (1-2 1 (1 (-,2) 1.2
2x2 + 1 h3 3.x (22 + 3) h4
+ + - - 2.3. + &c.
(1-)1 3 1 2.34
58. By the same means it is required to prove that
1 h 2x h2
tan-' ( + h) = tan- x + 2
1 +xI (1 qx)~1.2
2 (32-1) /3 2.3. 4x (2- 1) I4. _ &cd
(1 + 2)3 1.2.3 (1 + 2)4 1.2.3.4
dnu
59. Given u = (a - bx)n, to find the value of"d-.
dxu
60. Given u = (I- -)", to find the value of ddlu
61. Given u = (2e + v2)m% to find the value of d
v, dn~
u u
62. Given u = (1 + -2) + 3"2), to find the value off x.
d oo
63. Given u=(1,b )', to find the value of --
1 dn~u
64. Given u - (a2s q- 2)m, to find the value of
~~1 d~~du
65. Given u = - m, to find the value of
65. Given u --- to find the value of -
( d dw t
66. Given u. =, to find the value of
(1 -- 2,- + s3x2) d+,n
E