What is the source of this derivative formula We have just proved the n-th derivative of $f(x)=e^{-x}p(x)$, where $p(x)$ is a polynomial.
We got:
$$f(x)^{(n)}= e^{-x}(-1)^n\sum_{k=0}^{n}\binom{n}{k}(-1)^kp^{(k)}$$
Even though I do not completely understand the proof, I wonder, if there is a more general formula for expressions similar to this or if it has a name, so that I can look it up.

A generalization is
$$
\frac{d^n}{dx^n}\big(u(x) v(x)\big) = \sum_{k=0}^n \binom{n}{k} u^{(k)}(x) v^{(n-k)}(x)
$$
Proved by induction using the product rule.
This is sometimes called the "Leibniz Rule".
