18. Verify that $f_{x y}=f_{y x}$ for the following function. \[ f(x, y)=e^{x+y+13} \] \[ \begin{array}{l} f_{x}=\square \\ f_{y}=\square \end{array} \] \[ f_{x y}=f_{y x}= \]
Solution
Step 1 ：Given the function \(f(x, y)=e^{x+y+13}\), we are asked to verify that the second partial derivatives of the function with respect to \(x\) and \(y\) are equal. This is known as the equality of mixed partials, which is a property of all functions that have continuous second partial derivatives.
Step 2 ：First, we find the first partial derivatives of the function with respect to \(x\) and \(y\), denoted as \(f_x\) and \(f_y\) respectively. We have \(f_x = e^{x + y + 13}\) and \(f_y = e^{x + y + 13}\).
Step 3 ：Next, we find the second partial derivatives of the function with respect to \(x\) and \(y\), denoted as \(f_{xy}\) and \(f_{yx}\) respectively. We have \(f_{xy} = e^{x + y + 13}\) and \(f_{yx} = e^{x + y + 13}\).
Step 4 ：Since \(f_{xy} = f_{yx}\), we have verified the equality of mixed partials for the function \(f(x, y)=e^{x+y+13}\).
Step 5 ：Final Answer: \(\boxed{f_{xy} = f_{yx} = e^{x+y+13}}\).
