Binomial coefficient, combination for function derivatives

Question

Using Binomial theorem we know that

$$ \sum^n_{k=0}C^n_k\,x^ky^{n-k} = (x+y)^n. $$

I am wondering if let us choose a smooth real functions $f(x)$, and we have something like

$$ \sum^n_{k=0}C^n_k f^{(k)}f^{(n-k)} = C^n_0ff^{(n)} + C^n_1f'f^{(n-1)} + \cdots, = ??? $$ where we denote $f', \cdots, f^{(n)}$ as its derivatives. Does this leads to some simple expression like $(x+y)^n$ from the first equation?

Answer 1

Yes. The general Leibniz rule says, for $f,g$ $n$-times differentiable, $$ (fg)^{(n)}(x)=\sum_{k=0}^n\binom{n}{k}f^{(k)}(x)g^{(n-k)}(x). $$ So your question is just the special case $g=f$.