91Ó°ÊÓ

The first step is to find the first order partial derivatives of the function. The partial derivative of \(z\) with respect to \(x\) is found by differentiating \(z\) with respect to \(x\) while treating \(y\) as a constant. Symbolically, \[\frac{\partial z}{\partial x}=2x -2y\]Similarly, the partial derivative of \(z\) with respect to \(y\) is found by differentiating \(z\) with respect to \(y\) while treating \(x\) as a constant. Symbolically, \[\frac{\partial z}{\partial y}=-2x+6y\]

The next step is to find the second order partial derivatives. The second partial derivatives with respect to \(x\) and \(y\) are found by differentiating the first order partial derivatives obtained in the previous step.Using the first order partial derivatives, compute the second order partial derivatives:The second order partial derivative of \(z\) with respect to \(x\) twice gives: \[\frac{\partial^{2} z}{\partial x^{2}}=2\]The second order partial derivative of \(z\) taken first with respect to \(x\), then \(y\), gives: \[\frac{\partial^{2} z}{\partial x\partial y}=-2\]The second order partial derivative of \(z\) taken first with respect to \(y\), then \(x\) gives:\[\frac{\partial^{2} z}{\partial y\partial x}=-2\]The second order partial derivative of \(z\) with respect to \(y\) twice gives: \[\frac{\partial^{2} z}{\partial y^{2}}=6\]

Clairaut's Theorem states that if all mixed second partial derivatives are continuous in a region, then the mixed partial derivatives are equal, regardless of the order of differentiation. In this case, \[\frac{\partial^{2} z}{\partial x\partial y} = \frac{\partial^{2} z}{\partial y\partial x} = -2\]This shows that the two mixed second partial derivatives are equal, confirming the validity of Clairaut’s theorem for this function.