Problem 49 - College Algebra and Calculus: An Applied Approach

Question

Find the four second partial derivatives. Observe that the second mixed partials are equal. $$ z=\frac{e^{2 x y}}{4 x} $$

Step-by-Step Solution

Answer

The four second order partial derivatives can be calculated through these steps: Calculate the first order partial derivatives, then take the derivative again to get the second order derivatives according to x and y separately. Lastly, find the mixed second order derivatives by taking each initial derivative according to the other variable. Compare the mixed derivatives to check if they are equal.

Step 1: Calculate First Order Partial Derivatives

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

Step 2: Compute Second Order Partial Derivatives

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.

Step 3: Calculate 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.