# 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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