An introduction to mathematics, by A. N. Whitehead.

DIFFERENTIAL CALCULUS 25
Now take another example within the region
of pure mathematics. Let us proceed to find
the rate of increase of the function x2 for
any value x of its argument. We have not
yet really defined what we mean by rate of
increase. We will try and grasp its meaning
in relation to this particular case. When x
increases to x+h, the function x2 increases to
(x+h)2; so that the total increase has been
(x +h)2-x2, due to an increase h in the argument. Hence throughout the interval x to
(x +h) the average increase of the function per
unit increase of the argument is (x+h) -x2
But
(x +h)2 =x2+2hx+h2,
and therefore
(x +h)2 -x2 hzx +h2 +h
h h
Thus 2x+h is the average increase of the
function x2 per unit increase in the argument,
the average being taken over by the interval
x to x+h. But 2x+h depends on h, the size
of the interval. We shall evidently get what
we want, namely the rate of increase at the
value x of the argument, by diminishing h
more and more. Hence in the limit when h