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Substitution in implicit function

This is the code to the implicit differentiaion of a function from the theoretical point of view

    var("w1, w2, a, p")
    EU(w1, w2, a, p)= p*w1^a+ (1-p)*w2^a
    var("dw1, dw2")
    V_w1 = diff(V, w1)
    V_w2 = diff(V, w2)
    # Differential
    dV = V_w1 * dw1 + V_w2 * dw2 
    show("dV ="+latex(dV))
    # Dérivée du premier ordre
    sol=solve(dV==0, dw2)

Now without to be obliged to rewrite all the commands, I would like to apply this procedure to the EU function or any other function. I have tried to use dV.substitute_function(V, EU) and (sol[0]/dw1).right_hand_side() but without success. I have an amplified problem for the second order derivative since I am nosure of my formulation due to the fact that one must substitute the first order derivative inside derivation.