Let $f(x)= x^x$. Then
$f^{(n)}(x)$ if $n=1$ is $f'(x)=x^x\,(\ln(x)+1)$.
If $n=2$, $f''(x)=x^x\,(\ln(x)+1)^2+x^{x-1}$.
If $n=3$, $f'''(x)=x^x\,(\ln(x)+1)^3+3\,x^{x-1}\ln(x)+3x^{x-1}-x^{x-2}$
and so forth.
I don't actually know if this is the proper notation to ask this question but ill give an example of where it can be applied, to additionally help people understand the purpose in asking.
for any polynomials such as $f(x)=x^2+x+1$, $f^{(n)}(x)$ as $n→∞$
If $n=1$, $f'(x)=2x+1$.
If $n=2$, $f''(x)=2$.
If $n=3$, $f'''(x)=0$ and for any $n>3$ after, $f^{(n)}(x)=0$.
I am looking for anyway to simplify the pattern that occurs as you repeatedly differentiate such as a series or sequence that utilizes $n$, or an overall simplification such as $0$, $∞$, $-∞$, or that its divergent. If divergent, which I believe is likely, would there be representative formulas for odd and even n's?
Odd derivatives in green, Even in red.