Find \(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial^{2} f}{\partial x^{2}}, \frac{\partial^{2} f}{\partial y^{2}}\) and \(\frac{\partial^{2} f}{\partial x \partial y}\) if \(f=(x-y)^{2}\).

To find the first order partial derivatives, we differentiate the function \(f=(x-y)^{2}\) with respect to \(x\) and with respect to \(y\). $$ \frac{\partial f}{\partial x} =2(x-y), $$ and $$ \frac{\partial f}{\partial y} =-2(x-y). $$

Now we need to differentiate the first order partial derivatives with respect to \(x\) and \(y\) again to find the second order partial derivatives. Differentiating \(\frac{\partial f}{\partial x}\) with respect to \(x\), we get: $$ \frac{\partial^{2} f}{\partial x^{2}} = 2. $$ Differentiating \(\frac{\partial f}{\partial x}\) with respect to \(y\), we get: $$ \frac{\partial^{2} f}{\partial x \partial y} =-\frac{\partial^{2} f}{\partial y \partial x}= -2. $$ Differentiating \(\frac{\partial f}{\partial y}\) with respect to \(y\), we get: $$ \frac{\partial^{2} f}{\partial y^{2}} = 2. $$ So, the first and second order partial derivatives of the function \(f=(x-y)^2\) are: $$ \frac{\partial f}{\partial x}=2(x-y), $$ $$ \frac{\partial f}{\partial y}=-2(x-y), $$ $$ \frac{\partial^{2} f}{\partial x^{2}}=2, $$ $$ \frac{\partial^{2} f}{\partial y^{2}}=2, $$ and $$ \frac{\partial^{2} f}{\partial x \partial y}=-2. $$