StylusFalling asleep the other night I was thinking about derivatives.
I can think of two functions that are their own derivatives: f(x) = 0 and f(x) = ex.
There's a pair of functions function such that each is its second derivative is itself: f(x) = e-x and g(x) = -e-x.
There's a group of functions such that its fourth derivative is itself, f(x) = sin(x), g(x) = cos(x), h(x) = -sin(x), k(x) = -cos(x)
Are there any functions with other numbers of derivative steps before they become equal to themselves again? 3, 5, 8, etc.
In the light of day, I think I see that we could define an "n-self-derivative function" by just setting the derivatives
f(0) = 1
f'(0) = -1/2
f''(0) = -1/2
f'''(x) = f(x) etc
and that defines a unique Maclurin series.
But .. are there functions of this form that would arise naturally?
Update: Yes, one such function is exp(-x/2)*cos(x*sqrt(3)/2). see the response from @Chip_Unicorn: https://im-in.space/@Chip_Unicorn/116529978344298695
Chip Unicorn (@[email protected])
@[email protected] Math stack exchange has the answer: https://math.stackexchange.com/questions/1080016/function-whose-third-derivative-is-itself The answers are interesting!
Dani Laura (they/she/he)
greatquux
MOULE 
Adam R. Wood