## Back to e

Above, we defined as the particular real number satisfying

which gave us when . From this expression, we have, as ,

or

This is one way to define . Another way to arrive at the same definition is to ask what logarithmic base gives that the derivative of is . We denote by .

Numerically, is a transcendental number (a type of irrational number3.5), so its decimal expansion never repeats. The initial decimal expansion of is given by3.6

Any number of digits can be computed from the formula by making sufficiently small.

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Derivatives of f(x)=a^x