91Ó°ÊÓ

Firstly, find the first order partial derivatives with respect to \(x\) and \(y\). They are calculated as: \[\frac{\partial z}{\partial x} = \frac{\partial}{\partial x} \left(\frac{e^{2 x y}}{4 x}\right)\]\[\frac{\partial z}{\partial y} = \frac{\partial}{\partial y} \left(\frac{e^{2 x y}}{4 x}\right)\]

Next, find the second order partial derivatives. We will apply the rules of differentiation again.\[\frac{\partial^2 z}{\partial x^2} = \frac{\partial}{\partial x} \left(\frac{\partial z}{\partial x}\right)\]\[\frac{\partial^2 z}{\partial y^2} = \frac{\partial}{\partial y} \left(\frac{\partial z}{\partial y}\right)\] Depending on the results of the first order derivatives, it might also be necessary to calculate second order mixed partial derivatives.

Lastly, find the second order mixed partial derivatives. These derivatives are calculated by differentiating the result from Step 1 with respect to a different variable than the one initially used. So for the \(x\) derivative from Step 1, calculate the derivative of the solution with respect to \(y\), and vice versa for the \(y\) derivative.\[ \frac{\partial^2 z}{\partial x \partial y} = \frac{\partial}{\partial x} \left(\frac{\partial z}{\partial y}\right)\]\[ \frac{\partial^2 z}{\partial y \partial x} = \frac{\partial}{\partial y} \left(\frac{\partial z}{\partial x}\right)\] As a check, observe if these two mixed partial derivatives are equal, as this should safely be the case for a function such as this, due to Schwarz's theorem.