Exercise
Given the differentiable function
z(x,y)=x\ln y-y\ln x
Prove the equation
z''_{xy}=z''_{yx}
Proof
We will use the chain rule to calculate the partial derivatives of z.
z'_x=\ln y-y\cdot\frac{1}{x}
z'_y=\frac{x}{y}-\ln x
We will calculate the second order derivatives.
z''_{xy}=\frac{1}{y}-\frac{1}{x}
z''_{yx}=\frac{1}{y}-\frac{1}{x}
Hence, we got
z''_{xy}=z''_{yx}
As required.
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