I have this function: $G(x, y, z) = G(x, y, g(x,y))$ and the equation $$G = 0$$
I want to reach a specific equation: $$\frac{\partial{G}}{\partial{y}} + \frac{\partial{G}}{\partial{g}}*\frac{\partial{g}}{\partial{y}} = 0$$
I can see that by taking the total derivative in respect to y on the starting equation takes me to this equation: $$\frac{dG}{dy} = \nabla{G} * \begin{pmatrix}\frac{dx}{dy} & \frac{dy}{dy} & \frac{dg}{dy}\end{pmatrix} = \begin{pmatrix}\frac{\partial{G}}{\partial{x}} & \frac{\partial{G}}{\partial{y}} & \frac{\partial{G}}{\partial{g}}\end{pmatrix} \cdot \begin{pmatrix}0 & 1 & \frac{\partial{g}}{\partial{y}}\end{pmatrix} = 0$$
$$\frac{\partial{G}}{\partial{y}} + \frac{\partial{G}}{\partial{g}}*\frac{\partial{g}}{\partial{y}} = 0$$
Another idea is to take the partial derivate: $$G = 0$$ $$\frac{\partial{G}}{\partial{y}}= 0$$ $$\frac{\partial{G}}{\partial{y}} + \frac{\partial{G}}{\partial{y}}= 0$$ $$\frac{\partial{G}}{\partial{y}} + \frac{\partial{G}}{\partial{y}} \frac{\partial{g}}{\partial{g}}= 0$$ And exchange the order of differentiation. $$\frac{\partial{G}}{\partial{y}} + \frac{\partial{G}}{\partial{g}} \frac{\partial{g}}{\partial{y}}= 0$$ Exchanging the order of differentiation seems wrong to me, but $$G = 0$$ $$\frac{\partial{G}}{\partial{y}}= 0$$ seems wrong as well. Can we not take the partial derivative of both sides on an equation?