Multivariable Chain Rule – Proving an equation of partial derivatives – Exercise 6460

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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