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Partial differential of composite function

Let $ h(x_1,x_2,...x_n)=g(f_1(x_1,x_2,...x_n),f_2(x_1,x_2,...x_n),....f_m(x_1,x_2,...x_n)) $
Suppose that g and f are differentiable functions. Then h is differentiable and partial derivative can be founded by $ dh(x)=dg(f(x)df(x)  $. Each differential is a linear application and is represented by the matrix of partial derivatives.For example $ J_h(x)=(\frac{\partial h}{\partial x_1},\frac{\partial h}{\partial x_2},....\frac{\partial h}{\partial x_n}) $
Following composition rule for matrix, we have

$$\frac{\partial h}{\partial x_i}=\sum_{k=1}^{k=m}\frac{\partial g}{\partial y_k}(f)\frac{\partial f_k}{\partial x_i}$$

This sum expresses the product of a line in $ J_g (f(x)) $ with a column in $ J_f(x) $
or more explicitly

$$\frac{\partial h}{\partial x_i}(x_1,x_2,...x_n)=<br />
\sum_{k=1}^{k=m}\frac{\partial g}{\partial y_k}(f_1(x_1,x_2,...x_n),f_2(x_1,x_2,...x_n),....f_m(x_1,x_2,...x_n))\frac{\partial f_k}{\partial x_i}(x_1,x_2,...x_n)$$

If g and f are twice differentiable second order partial differential can be express

$$\frac{\partial^2 h}{\partial {x_i}_2\partial {x_i}_1}=\sum_{k_2=1}^{k_2=m}\sum_{k_1=1}^{k_1=m}\frac{\partial ^2 g}{\partial {y_k}_2\partial {y_k}_1}(f)\frac{\partial {f_k}_2}{\partial {x_i}_2}\frac{\partial {f_k}_1}{\partial {x_i}_1}+\sum_{k_1=1}^{k_1=m}\frac{\partial g}{\partial {y_k}_1}(f)\frac{\partial ^2{f_k}_1}{\partial {x_i}_2\partial {x_i}_1}$$
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