Deriving a high order Derivative formula for fractional variable

Let $f\in C^{(\infty)}(\Bbb{R})$. Show that for $x\ne 0$:$$\frac{1}{x^{n+1}}f^{(n)}\left(\frac 1x\right)=(-1)^n\frac{d^n}{dx^n}\left(x^{n-1}f\left(\frac 1x\right)\right)$$

I think a possible approach to prove this is by induction, but I haven't found a way through the derivation. Any help is greatly appreciated, thank you.