Derivatives of f(x)=a^x
Let's apply the definition of differentiation and see what happens:

Since the limit of
as
is less than
1 for
and greater than
for
(as one can show via direct
calculations), and since
is a continuous
function of
for
, it follows that there exists a
positive real number we'll call
such that for
we get



So far we have proved that the derivative of is
.
What about
for other values of
? The trick is to write it as












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