Dynneson
The Euler Circular-Reasoning Gap: The Exponential Revisited
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ABSTRACT.  Realizing that the derivatives of the exponential and logarithmic functions are invariably circular in their reasoning, what I set out to do was to attempt to "close the loop," by expanding the intuition to a level of abstraction not normally achieved at the Calculus I level. In fact, Euler's Formula is usually reserved for Calculus II at the earliest, and can be attained as an example of Taylor's Expansion. Instead, in the following discourse, Euler's Formula is derived by way of exponential-growth. Passing to the complex-realm and applying DeMoivre's Theorem, it inevitably becomes circular.