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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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