One step in the derivation of Black-Scholes
Assumptions:(1) ${\displaystyle \frac{\partial F}{\partial t}(t,x)+\frac{1}{2}\sigma^{2}x^{2}\frac{\partial^{2}F}{\partial x^{2}}(t,x)-rF(t,x)+rx\frac{\partial F}{\partial x}(t,x)=0}$
(2) $\tilde{F}(t,x)=e^{-rt}F(t,xe^{rt})$ .
Show that ${\displaystyle \frac{\partial\tilde{F}}{\partial t}(t,x)=-\frac{1}{2}\sigma^{2}x^{2}\frac{\partial^{2}\tilde{F}}{\partial x^{2}}(t,x)}$.
Although to derive Black-Scholes we might need PDE and Ito's lemma, I think here we only need calculus. However I cannot seem to get it right. I guess my problem has something to do with the partial derivative of $F(t,xe^{rt})$. Thank you!