Say we have some function $\psi'(r,t)$, given by $\psi'(r,t)=e^{af(r,t)}\psi(r,t)$
and we want to calculate $\nabla^2\psi'(r,t)$.
I obviously know how to do all the basic steps of this product rule , chain rule and all that but there's one part I'm a little unsure of ( I always found it a little confusing , I used to know how to do it but I forgot )
Basically in the parts of the equation where we have $\tfrac{\partial f(r,t)}{\partial r},\tfrac{\partial f(r,t) }{\partial t}, \tfrac{\partial \psi(r,t)}{\partial r},\tfrac{\partial\psi(r,t)}{\partial t}$, I want to know how we can deal with these ?
(If what I'm asking isn't clear , please ask me to elucidate and I'll write more about what it is specifically that confuses me )
WLOG, we can parameterize r and t by some parameter $\tau$. We then have $$\frac{d f}{d \tau} = \frac{\partial f}{\partial r}\frac{dr}{d\tau}+\frac{\partial f}{\partial t}\frac{dt}{d\tau}$$
From here, you should be able to re-arrange the equation and use identities similar to the Maxwell relations to have your formula depend only on full derivatives instead of partials, which in my opinion, would be a simplified version.