Proof of the derivative of an elementary function is also elementary

The closure under differentiation can be proven by induction.

The result is clear for « base functions » ($\ln x, e^x)...$

If a map is a sum, the product, the composition... of two simple functions, then it follows that its derivative is a simple function based on the induction hypothesis and the differentiation rules of the sum, the product, the composition... of two functions.

As any simple function can be obtained by induction using previous rules according to the very definition of a simple function, we’re done.

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Elementary Functions

Proof Explanation

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