Here is the code for first and second order implicit differentiation of a two variables function

#Evaluation of the two first derivatives of an implicit function
var("x, y, dx, dy, al, be")
V=function('V')(x, y)# A) either A or B should be uncomment

# V= x^al*y^be# B)

V_x = diff(V, x)
V_y = diff(V, y)

# Evaluation of the two first derivatives of an implicit function

var("x, y, dx, dy, al, be")
V=function('V')(x, y)# A) either A or B should be uncomment

# V= x^al*y^be# B)

V_x = diff(V, x)
V_y = diff(V, y)

# Differential

dV = V_x * dx + V_y * dy
show(dV)

```
# Dérivée du premier ordre
sol=solve(dV==0, dy)
show(sol[0]/dx)
# Dérivée du second ordre
y_x=sol[0].rhs()/dx
show(y_x)
hh=y_x.function(x,y)
y=function('y')(x)
hh_x=diff(hh(x,y),x).full_simplify().subs(diff(y(x), x)==y_x).full_simplify()
show(hh)
show(hh_x)
```

$\frac{\mathit{dy}}{\mathit{dx}} = -\frac{\frac{\partial}{\partial x}V\left(x, y\right)}{\frac{\partial}{\partial y}V\left(x, y\right)}$

$-\frac{\frac{\partial}{\partial x}V\left(x, y\right)}{\frac{\partial}{\partial y}V\left(x, y\right)}$

Now I will work on the substitution of the chosen function. And when I will have found I will come back.