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

RULES FOR DIFFERENTIATION.

Rule VI. If y = logx, dy= d
X
Rule VII1. If y = ax, dy =log a.a~vdx.

Rule VIII.

If y = sinx,
If y =cos X,

dy = cos dx.
dy = - sin xdx.

Rule IX. If y = tan x,
If y = cot X,
RIule X. If y = sin' a
If y =cos -'a,Rule XI. If y = tanQ,
if y = cot-' C

dx
dy = ~ x
dr
dy =d
dx
d, dy =2
dx
4 -~/ -m
r, dy=1 + 'F2

Rule XII. 90. Rule XII. shews that if we obtain dx in terms of dy
from the equation x = j(y), the value of so obtained is the
dx
proper value; i. e. the same value that would be obtained if
we found y in terms of x directly, and then differentiated. For
by differentiating the equation x q 5(y) as it stands, we find
dy 1
dx = qY(y)dy, and dx
dx 5'(y)
Now if f(x) be the value of y found directly in terms of x, we
dy
have i =f' (x); which is the same as the former value, since
X2L