This may well be a stupid question. I'm currently trying to find out whether a production function I have has convex isoquants.
I'm aware I can find the derivative $\frac{dL}{dK}$ by using the rule of implicit differentiation:
$$\frac{dL}{dK} = -\frac{∂Q}{∂K}/\frac{∂Q}{∂L} $$
Where Q is the production function.
Is it possible to find the second derivative in a similar way? i.e.
$$\frac{d^2L}{dK^2} = -\frac{∂^2Q}{∂K^2}/\frac{∂^2Q}{∂L^2} $$
Hint: The equation is not the result of implicit differentiation. It is the result of total differentiation:
$$dQ(K,L)=\frac{\partial Q}{\partial L}\cdot dL+\frac{\partial Q}{\partial K}\cdot dK=0$$
$$\frac{\partial Q}{\partial L}\cdot dL=-\frac{\partial Q}{\partial K}\cdot dK$$
Dividing the equation by $\frac{\partial Q}{\partial L}$ and $dK$ gives the final result.