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If z is a function of x and y, then:$$dz=\frac{\partial z}{\partial x}dx+\frac{\partial z}{\partial y}dy$$So, $M=\frac{\partial z}{\partial x}$ and $N=\frac{\partial z}{\partial y}$. Since the order of taking partial derivatives is immaterial,$$\frac{\partial^2 z}{\partial x \partial y}=\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$$



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