What's your favorite illustration of the usefulness of complex numbers?
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I am not stopping to appeal to the Moivre formule
. But when you derive in both sides of the formule, you already know the derive of the Sin(x) and Cos(x), and after you use it to “prove” the derive of Sin(x) and Cos(x). Its a circular reasoning that obviously is invalid.
?? i did not even use the derivative of sin(x) nor cos(x) in this at all.. that's the whole point ..
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look closer .. the only thing that was used was d/dx e^(a x) = a e^(a x)
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