Say I have a function
$$ f(x) = \dfrac 1x$$
and I'm looking at its $n^{th}$ derivative and trying to come up with a formula.
I can easily get it because if forms a very consistent pattern and it somewhat reminds me of harmonic series.
The formula is
$$ f'^n (x) = (-1)^n {n! \over n^{n+1}}$$
However, for the following example I can't come up with a general formula
$$ f(x) = e^x \cos(x)$$
I calculated the first 5 derivatives for it and there's a pattern that repeats every subsequent $4^{th}$ derivative
$f'(x) = e^x (\cos x -\sin x)$
$f''(x) = -2e^x (\sin x)$
$f'''(x) = -2e^x (\cos x +\sin x)$
$f''''(x) = -4e^x (\cos x)$
So, there's an alternating sign pattern and the $\sin x$ and $\cos x$ patterns which I just can't seem to account for in a general formula for the $n^{th}$ derivative.
Any help would go a long way!