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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36. If y=ae + be- + c sin (x +e), then will s - y =o,
d4y
if s - -.
37. If y = ew (a + bx) + c sin (x + e), then will

38.
a,, /3 7'
39.
nate c.

-s- 2r +2q —2p+y=o.
If y = ae"" + be~ + ce' l + &c. to n terms, where
&c. are the n roots. of 1, then will d - = o.
If +(-c)(y+ c, it is required to elimi
If yx + (1 - c) (y + x) =- c, it is required to elimi

40. Eliminate a and b from. the equation
ax2 + by2 = ax + by.
41. Eliminate sin w and sin 2 from the equation
y = a sin x + b sin 2..
C
42. From y=aek+ bel - - cos nx, it is required to
qmn
eliminate the terms involving a and b, when k and I are the
roots of the equation z2 + mz + n2 = o.
43. Eliminate the constant quantities and exponential
function from the equation y = e' (a + bw + cx2 + &c. + IZ"'1).
III. Developement of Functions, Sc. as in Articles (69)-(86).
1. If = (a — X)4, then u= a4 -4ax+ 3 6a2'' -4a + x4.
2. If u= (a2+-2)5, then will
u = al'T+ -8 5asx2 l10a6x4 + 100a4xO 6 + 5a2XS8 + lo,1
3. If u= 2/1 +, then will
1 1 <'2 3 X3 3.5 x4
u=l + -x- --- + &c.
2 2" 1.2 2 1.2.3 24 1.2.3.4
4. If u = a2 (1 - 2), then will
a2 62X 6 a2 X 6. 1 1 a2 6. 1 1. 16ax
U-aI?+, - + ---- + ------ + ------ +.&C.
5 5.10 5.10.15 5.10.15. -0