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

Viii SUMMARY.
ARTICLE PAG
23. If we have it = p - q + r - &c., wherein p, q, r, &c. are
all functions of the same independent variable x, then will
du dP dq dr
dx dx d-Tx dx
and du=dp-dq+dr-&c................ 26
24. If we have it pq, p and q being functions of x, then will
du dp dq
W- + p$P, and du = qdp + pdq........ 27
27. If u =V, where p and q are both functions of x, then will
q
du 1 dp dq qdp -dpdq
dx dx dx q2.. 29
31. If u = ptm where p is any function whatever of x, then will
du 1dp)
dir nMp M-1 d- -, and du = mpm-ldp,
whether m be positive or negative, integral or fractional. 30
CHAP. IIJ.
Differentiation of Exponential and Logarithmical Function8 of one
principal Variable.
34. If it = aP, where p is any function whatever of x, then will
du dp
=:0 log a aP and d-l = log a aPdp,
dx dx
where the logarithm of a is taken in the system whose base
is 2.71828 &c. represented by e...........................- 38
9. If u =t log p, where p is any function of x, then will
du _ dp_ d
- M.f -,and du=M
dx pdx p
where M is the modulus of the system of logarithms used.. 40
42. If it = p9, where p and q are both functions of the same independent variable x, then will
du = dq dp an p Sb d+ qldp 41
dl plog p W- + qpg- jx-, and du p opdq+ p4
44. If ut = log of log p, which is usually written nt = log2p, then
will dplogpdx anddu= 1dp...42
dx - plogp dx' 10. P.....