Assume that
and that
.
The integral of
is
, where
is an arbitrary constant. The integral of
is
, where again
is an arbitrary constant and
is the natural logarithm of
, often written as
.
When
is close to zero,
and
are close, so there must be some connection between their integrals!
Choose
and
so that the two integrals are both zero at
. The integrals are then
and
. For
close to zero these functions are very close; in symbols,
.
Using the difference quotient for the derivative of the base-
exponential function
with respect to
(not
) and using
instead of the more usual
gives
. This is more usually written with
as the variable:
, with the special case
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