Simply put if I follow the rule derived by the simple proof denoting $e^x$ to be the derivative of $e^x$ then it follows that it should have an infinite number of derivatives. Is this a conceptual mistake ( on my part) or one deeper within the proof ?
2026-05-14 03:20:22.1778728822
How can one properly understand the fact that $e^x$ can be differentiated an infinite amount of times?
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There is no mistake, with your remark, a trivial induction gives you the existence and value of the n-th derivative for every n, and this imply that it's $C^\infty$ – Tryss [answered in comments]