Multivariable Chain Rule – Proving an equation of partial derivatives – Exercise 6462

Exercise

Given the differentiable function

$z(x,y)=x^y$

Prove the equation

$z''_{xy}=z''_{yx}$

Proof

We will use the chain rule to calculate the partial derivatives of z.

$z'_x=yx^{y-1}$

$z'_y=x^y\ln x$

We will calculate the second order derivatives.

$z''_{xy}=x^{y-1}+y\cdot\frac{1}{x}\cdot x^y\cdot\ln x=$

$=x^{y-1}(1+y\cdot\ln x)$

$z''_{yx}=yx^{y-1}\cdot\ln x+x^y\cdot\frac{1}{x}$

$=x^{y-1}(y\cdot\ln x+1)$

Hence, we got

$z''_{xy}=z''_{yx}$

As required.

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