I haven't done this sort of thing "in anger" for over 25 years, but I thought I'd have a go just to see if the old magic was still there as I used to enjoy this sort of methods-type question in my braver days
Apart from one false start and a stupid inability to replace n with n+1 consistently I managed to get it out reasonably quickly, but I guess you need to pick the right strategy otherwise it's easy to go round in circles!
I started by differentiating the 1st eq w.r.t.x and then using the 2nd eq to substitute for
Pn+1′This gives
Pn+1′ in terms of
Pn and
Pn′.
Multiplying this new eq by x and comparing with the original 2nd eq with n replaced by n+1 then gives
Pn′ in terms of
Pn+1 and
Pn.
Differentiation w.r.t.x gives an equation in
Pn′′ and
Pn+1′ and the latter can be substituted using one of our earlier results.
A bit of simple algebra then gives the required result