Going backwards from derivative value, to retrieve point and order.

Question

Using Sage let's say I do the following.

Take the $4^{th}$ order derivative of $\sin(x)$ about the point $2$:

sage: from mpmath import mp
sage: a = mp.diff(lambda x: sin(x), 2, 4)
sage: a
mpf('0.909297426825681...')

Now how do I go from the value computed above back to the point (value of $2$) and $n^{th}$ order (value of $4$).

I'm not sure if this involves using the integral function.

Thanks in advance!

Answer 1

The computed value does not remember what computation it came from.

This floating point number a = 0.909... is just a floating-point number.

Just from a, no way to know if it was computed as $\sin(2)$, $\sin^{(4)}(2)$, $\sin^{(8)}(2)$, $\sin(2 + 2\pi)$, $\cos(2-\pi/2)$, ... or if it was just input by typing all its digits.