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

For

, we thus have

.
So far we have proved that the derivative of

is

.
What about

for other values of

? The trick is to write it as

and use the

chain rule,

^{3.3} where

denotes
the log-base-

of

.

^{3.4} Formally, the chain rule tells us how to
differentiate a function of a function as follows:

Evaluated at a particular point

, we obtain

In this case,

so that

,
and

which is its own derivative. The end result is then

,

*i.e.*,

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