Cake eating problem
Observe that the partial derivative is by the parameter of the goal function
so with
g(p,x) = βT ^2 u(x)+βT ^1 u(p x)
∂g
∂p(p,x(p)) = β
T (^1) u (^0) (p x(p))
that is
dVT 1 (w)
dw (w)=β
T (^1) u (^0) (w c(w)).
Which gives the Euler equation.