Given the function $f(x,y) = x^2y + xe^y$, let's find the second-order partial derivatives $f_{xx}(x,y)$, $f_{xy}(x,y)$, $f_{yx}(x,y)$, and $f_{yy}(x,y)$.

1. First-order partial derivatives:

   • With respect to $x$: $f_x(x,y) = \frac{\partial}{\partial x}(x^2y + xe^y) = 2xy + e^y$
   • With respect to $y$: $f_y(x,y) = \frac{\partial}{\partial y}(x^2y + xe^y) = x^2 + x e^y$

2. Second-order partial derivatives:

   • With respect to $x$ twice: $f_{xx}(x,y) = \frac{\partial}{\partial x}(2xy + e^y) = 2y$ So, $A = 2$.

   • With respect to $x$ then $y$: $f_{xy}(x,y) = \frac{\partial}{\partial y}(2xy + e^y) = 2x + 0 = 2x$

   • With respect to $y$ then $x$: $f_{yx}(x,y) = \frac{\partial}{\partial x}(x^2 + xe^y) = 2xy + e^y = 2x + e^y$

   • With respect to $y$ twice: $f_{yy}(x,y) = \frac{\partial}{\partial y}(x^2 + xe^y) = 0 + xe^y = xe^y$ So, $D = x$.

Combining these, we get:

• $f_{xx}(x,y) = 2y$ (with $A = 2$),
• $f_{xy}(x,y) = f_{yx}(x,y) = 2x + e^y$ (with $B = 2$ and $C = 1$),
• $f_{yy}(x,y) = xe^y$ (with $D = x$).

So the values are:

• $A = 2$,
• $B = 2$,
• $C = 1$,
• $D = x$.

Would you like more details on any of these steps?

Suggested Questions:

1. How do you find higher-order partial derivatives for more complex functions?
2. What are mixed partial derivatives, and how do they relate to each other?
3. Can you explain the concept of symmetry in second-order mixed partial derivatives?
4. How are partial derivatives used in optimizing functions with multiple variables?
5. What are some common applications of second-order partial derivatives in real-world problems?

Tip:

When computing second-order partial derivatives, ensure consistency by checking if the mixed partial derivatives $f_{xy}$ and $f_{yx}$ are equal, which they should be if the function and its partial derivatives are continuous.