Find $f_{x x}, f_{x y}, f_{y x}$, and $f_{y y}$ for the following function. (Remember, $f_{y x}$ means to differentiate with respect to $y$ and then with respect to $x$.) \[ f(x, y)=e^{6 x y} \] \[ \mathrm{f}_{\mathrm{xx}}= \] \[ f_{x y}= \] \[ f_{y x}= \] \[ f_{y y}= \]
Solution
Step 1 ：Given the function \(f(x, y)=e^{6 x y}\), we are asked to find the second order partial derivatives \(f_{x x}, f_{x y}, f_{y x}, f_{y y}\).
Step 2 ：First, we find the first order partial derivatives: \(f_x = \frac{\partial f}{\partial x} = 6ye^{6xy}\) and \(f_y = \frac{\partial f}{\partial y} = 6xe^{6xy}\).
Step 3 ：Next, we find the second order partial derivatives: \(f_{xx} = \frac{\partial^2 f}{\partial x^2} = \frac{\partial f_x}{\partial x}\), \(f_{xy} = \frac{\partial^2 f}{\partial x \partial y} = \frac{\partial f_x}{\partial y}\), \(f_{yx} = \frac{\partial^2 f}{\partial y \partial x} = \frac{\partial f_y}{\partial x}\), and \(f_{yy} = \frac{\partial^2 f}{\partial y^2} = \frac{\partial f_y}{\partial y}\).
Step 4 ：Calculating these derivatives, we find: \(f_{xx} = 36y^2e^{6xy}\), \(f_{xy} = 36xye^{6xy} + 6e^{6xy}\), \(f_{yx} = 36xye^{6xy} + 6e^{6xy}\), and \(f_{yy} = 36x^2e^{6xy}\).
Step 5 ：Thus, the second order partial derivatives of the function \(f(x, y)=e^{6 x y}\) are: \(\boxed{f_{x x} = 36y^2e^{6xy}}\), \(\boxed{f_{x y} = 36xye^{6xy} + 6e^{6xy}}\), \(\boxed{f_{y x} = 36xye^{6xy} + 6e^{6xy}}\), and \(\boxed{f_{y y} = 36x^2e^{6xy}}\).
