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

42

RULES FOR FINDING f' (X).

Let f(x) = X x; then logf(t) = xlogex,
and therefore we have.f' () 1
- = log o +. -;
f () v'.. f' () = a (logx + 1).
ft 77. * To these Rules we must add two more, which
will be found of great use in many cases, and important hereafter, when we come to consider more than one independant
variable. But we must first explain a certain notation which
we shall make use of.
Partial de- If f (vy) be any function of two variables x and y, we
rivatives;
what. The shall assume f'(wy) to denote the limiting value of
notation
q,'(ay) and Y) when v' approaches v. It is evident that
0P'(xy). Q' - iv
this limiting value is the derived function of f(xy), taken
on the supposition that y is constant and x alone variable.
It is called the Partial Derivative off(xy) with respect to x.
In the same manner we shall take f'(Xy) to denote the
limiting value of f(y') -,f(,) when y' approaches y;
y -y
i. e. the Partial Derivative of f (xy) with respect to y, or the
derivative taken on the supposition that y alone is variable.
Examples. Thus if f (Xy) = 'v2 + iy + y2, then
f'(xy) = 2x + y, and f'(.:y) = y + 2i.
Again, if f(xy) = x sin y + yet,
f' (y) = sin y + y e, and f' (x) = x cos y + e".
Thus by dashing the functional letter we denote a derivative of a function of two variables, and by a dot under
one of the variables we signify that that variable alone is
supposed to vary.
This nota- In using this notation we do not necessarily assume
tion does
not neces- that y is independant of x; for whether y be independsarily require that * The Articles marked thus tt - may be omitted till they are referred to, at least
on a first perusal.