i actually thought the opposite. i found it extremely illuminating to see exp(ix) partition the power series into even and odd terms and collect into cos and sine.

i just always thought of the chain of functions that are their own derivatives.

f = f' => f = exp

f = f'' => f = cosh or sinh

f = f'''' = -f'' => f = cos or sine

if you dont skip cosh/sinh the pattern makes more sense.

and obviously f = f'''' strongly suggests 1 = i^4