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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28. If (x + y + z) ez = c, prove that by eliminating, the
constant quantity and exponential function we shall have
dx +dyt+dz + (x +y +,) dzO=.
29. if z -F y t = (y + z), it is required to prove that
(yz +z2) dx + zt- xz) dy +(xy -yt) dz-+ (y2 +yz) dt =.
30. If u =f (z) and z = (x, y), it is required to prove
du du_ dz dz
that -- -; and thence to shew that the indedx 'dy dx dy
terminate function f of a determinate function z of two variables x and y may be eliminated.
31. If the indeterminate function be eliminated from the
x
equation z - i-f (ay - bx), it is required to prove that
a
dz bdz
dx dy
32. If z'= then will px +qy O, where
P zand q =
33. If z =aVX2~2+j (f) then pv +qy ='a 'V2 y2.
34. Ifz=xy+P(k ) x thenwillpxz+qyz=xy.
35. IfZ= ~ ( +logx), then will ap -qx =.
a
36. If z=- 1ogy+44 x,, then willnpx +mqy=a37 IfZ (X + y)mp95(X2 _y2), then will py +qx m%.
2
38. Ifx S+, y), then Will qXy =pX2 +y2.
3X