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I have an implicitly defined $C^1$ function $z=g(x, y)$ as a solution of $z$ in terms of $x$ and $y$ to $G(x, y, z)=0$ of some $C^1$ function $G$.

Then I have the derivative of $g$ $$\frac{\partial G}{\partial (x, y)}(a, b)+\frac{\partial G}{\partial z}(a, b)\cdot Dg(a, b)=0$$

I'm wondering how to get the second order partial derivatives $D_1D_2g, D_1D_1g, D_2D_2g, D_2D_1g$ from this.

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something like this $$ \frac{\partial}{\partial x}\left(\frac{\partial G}{\partial (x, y)}(a, b)\right)+\frac{\partial}{\partial x}\left(\frac{\partial G}{\partial z}(a, b)\cdot Dg(a, b)\right)=0$$ and then use chain rule, similarly in second direction

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