What's your favorite illustration of the usefulness of complex numbers?
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To be fair, it looks like the actual proof of those would be seen in the first equality of the second line of math. It's a good trick to remember the derivatives, but not really a proof of their values.
this constitutes a proof under the assumptions (e^(ax))' = a*e^(ax) and e^(ix) = cos(x) + i*sin(x)
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Right, but prove that first equation (over complex numbers), and you'll likely see the derivatives pop out in the process. Depending on the method of proof, of course. I couldn't see the earlier slides from your talk.
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it follows straight from the limit definition
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and that the derivative of sin and cos is real-valued
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