differentiation

$J f = ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎡ \frac{\partial f _{0}}{\partial x _{0}} \frac{\partial f _{1}}{\partial x _{0}} ⋮ \frac{\partial f _{m - 1}}{\partial x _{0}} \frac{\partial f _{0}}{\partial x _{1}} \frac{\partial f _{1}}{\partial x _{1}} ⋮ \frac{\partial f _{m - 1}}{\partial x _{1}} \dots \dots ⋱ \dots \frac{\partial f _{0}}{\partial x _{n - 1}} \frac{\partial f _{1}}{\partial x _{n - 1}} ⋮ \frac{\partial f _{m - 1}}{\partial x _{n - 1}} ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎤$

$\frac{\partial}{\partial x} (f \cdot g) = \frac{\partial f}{\partial x} \cdot g + f \cdot \frac{\partial g}{\partial x}$

$\frac{\partial}{\partial x} (f + g) = \frac{\partial f}{\partial x} + \frac{\partial g}{\partial x}$

$\frac{\partial}{\partial x} (f \circ g) = \frac{\partial f}{\partial g} \frac{\partial g}{\partial x}$

$\frac{\partial}{\partial x} (f \circ g) = \frac{\partial f}{\partial g} \frac{\partial g}{\partial x} = [\frac{\partial f}{\partial g _{1}} \frac{\partial f}{\partial g _{2}}] [\frac{\partial g _{1}}{\partial x} \frac{\partial g _{2}}{\partial x}]^{T} = \frac{\partial f}{\partial g _{1}} \frac{\partial g _{1}}{\partial x} + \frac{\partial f}{\partial g _{2}} \frac{\partial g _{2}}{\partial x}$

$f (x_{1}, x_{2}) = x_{1} (x_{1} + x_{2})$

$g_{1} = g_{2} = g_{3} = g_{4} = x_{1} x_{2} g_{1} + g_{2} g_{1} \cdot g_{3}$

$\frac{\partial f}{\partial x _{1}} = \frac{\partial g _{4}}{\partial x _{1}} = \frac{\partial}{\partial x _{1}} (g_{1} \cdot g_{3}) = \frac{\partial g _{1}}{\partial x _{1}} g_{3} + g_{1} \frac{\partial g _{3}}{\partial x _{1}} = \frac{\partial g _{1}}{\partial x _{1}} g_{3} + (g_{1} \frac{\partial g _{1}}{\partial x _{1}} + g_{1} \frac{\partial g _{2}}{\partial x _{1}})$

$\frac{\partial f}{\partial x _{1}} = \frac{\partial f}{\partial g _{4}} \frac{\partial g _{4}}{\partial x _{1}} = \frac{\partial g _{4}}{\partial g _{1}} \frac{\partial g _{1}}{\partial x _{1}} + \frac{\partial g _{4}}{\partial g _{3}} \frac{\partial g _{3}}{\partial x _{1}} = \frac{\partial g _{4}}{\partial g _{1}} \frac{\partial g _{1}}{\partial x _{1}} + \frac{\partial g _{4}}{\partial g _{3}} (\frac{\partial g _{3}}{\partial g _{1}} \frac{\partial g _{1}}{\partial x _{1}} + \frac{\partial g _{3}}{\partial g _{2}} \frac{\partial g _{2}}{\partial x _{1}})$

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differentiation