Suppose $f\in C^{\infty}(\mathbb R)$. Show that for $x \neq 0$ $$\frac{1}{x^{n+1}}f^{(n)}(\frac{1}{x})=(-1)^{n}\frac{d^n}{dx^n}[x^{n-1}f(\frac{1}{x})]$$
I am supposed to attempt it from right-hand side to left-hand side using Leibniz product rule for derivative, but I found out that I stuck on simplifying the sigma notation. Any clue for it?