2 dimensional case

In this case we may write the name of the variable (x,y)
let $ h(x,y)=g(a(x,y),b(x,y)) $
Then

$$\frac{\partial h}{\partial x}=\frac{\partial g}{\partial a}(a,b)\frac{\partial a}{\partial x}+\frac{\partial g}{\partial b}(a,b)\frac{\partial b}{\partial x}$$
$$\frac{\partial h}{\partial y}=\frac{\partial g}{\partial a}(a,b)\frac{\partial a}{\partial y}+\frac{\partial g}{\partial b}(a,b)\frac{\partial b}{\partial y}$$

$ \frac{\partial ^2h}{\partial x^2}=<br />
\frac{\partial ^2g}{\partial a^2}(a,b){\frac{\partial a}{\partial x}}{\frac{\partial a}{\partial x}}+<br />
\frac{\partial ^2g}{\partial b\partial a}(a,b)\frac{\partial b}{\partial x}\frac{\partial a}{\partial x}+<br />
\frac{\partial ^2g}{\partial a\partial b}(a,b)\frac{\partial a}{\partial x}\frac{\partial b}{\partial x}+<br />
\frac{\partial ^2g}{\partial b^2}(a,b)\frac{\partial b}{\partial x}\frac{\partial b}{\partial x}+<br />
\frac{\partial g}{\partial a}(a,b)\frac{\partial ^2 a}{\partial ^2x}+\frac{\partial g}{\partial b}(a,b)\frac{\partial ^2b}{\partial x^2} $

$ \frac{\partial ^2h}{\partial x\partial y}=<br />
\frac{\partial ^2g}{\partial a^2}(a,b){\frac{\partial a}{\partial x}}{\frac{\partial a}{\partial y}}+<br />
\frac{\partial ^2g}{\partial b\partial a}(a,b)\frac{\partial b}{\partial x}\frac{\partial a}{\partial y}+<br />
\frac{\partial ^2g}{\partial a\partial b}(a,b)\frac{\partial a}{\partial x}\frac{\partial b}{\partial y}+<br />
\frac{\partial ^2g}{\partial b^2}(a,b)\frac{\partial b}{\partial x}\frac{\partial b}{\partial y}+<br />
\frac{\partial g}{\partial a}(a,b)\frac{\partial ^2 a}{\partial x\partial y}+\frac{\partial g}{\partial b}(a,b)\frac{\partial ^2b}{\partial x\partial y} $

$ \frac{\partial ^2h}{\partial y^2}=<br />
\frac{\partial ^2g}{\partial a^2}(a,b){\frac{\partial a}{\partial y}}{\frac{\partial a}{\partial y}}+<br />
\frac{\partial ^2g}{\partial b\partial a}(a,b)\frac{\partial b}{\partial y}\frac{\partial a}{\partial y}+<br />
\frac{\partial ^2g}{\partial a\partial b}(a,b)\frac{\partial a}{\partial y}\frac{\partial b}{\partial y}+<br />
\frac{\partial ^2g}{\partial b^2}(a,b)\frac{\partial b}{\partial y}\frac{\partial b}{\partial y}+<br />
\frac{\partial g}{\partial a}(a,b)\frac{\partial ^2 a}{\partial ^2y}+\frac{\partial g}{\partial b}(a,b)\frac{\partial ^2b}{\partial y^2} $

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