The problem asks us to find the slope of the surface $z = f(x, y) = \ln(xy^2 + 2xy) + xy$ in the $x$-direction at the point $(2, -1)$.

Step 1: Partial derivative with respect to $x$

To find the slope in the $x$-direction, we need to compute the partial derivative of the function $z = f(x, y)$ with respect to $x$.

The function is: $z = \ln(xy^2 + 2xy) + xy$

First term: $\ln(xy^2 + 2xy)$

Using the chain rule: $\frac{\partial}{\partial x} \ln(xy^2 + 2xy) = \frac{1}{xy^2 + 2xy} \cdot \frac{\partial}{\partial x}(xy^2 + 2xy)$ We now compute $\frac{\partial}{\partial x}(xy^2 + 2xy)$: $\frac{\partial}{\partial x}(xy^2 + 2xy) = y^2 + 2y$ So, $\frac{\partial}{\partial x} \ln(xy^2 + 2xy) = \frac{y^2 + 2y}{xy^2 + 2xy}$

Second term: $xy$

The partial derivative of $xy$ with respect to $x$ is: $\frac{\partial}{\partial x}(xy) = y$

Step 2: Total partial derivative with respect to $x$

Thus, the partial derivative of $z$ with respect to $x$ is: $\frac{\partial z}{\partial x} = \frac{y^2 + 2y}{xy^2 + 2xy} + y$

Step 3: Evaluate at the point $(2, -1)$

Now, we substitute $x = 2$ and $y = -1$ into the partial derivative: $\frac{\partial z}{\partial x} = \frac{(-1)^2 + 2(-1)}{2(-1)^2 + 2(2)(-1)} + (-1)$

Simplifying the terms: $\frac{\partial z}{\partial x} = \frac{1 - 2}{2(1) + 2(-2)} + (-1) = \frac{-1}{2 - 4} - 1 = \frac{-1}{-2} - 1 = \frac{1}{2} - 1 = -\frac{1}{2}$

Final Answer:

The slope of the surface in the $x$-direction at the point $(2, -1)$ is: $-\frac{1}{2}$

Would you like further clarification or details on this process? Here are some related questions:

1. How do you find partial derivatives of more complex functions involving logarithms?
2. What is the physical interpretation of the partial derivative in the $x$-direction?
3. How does the chain rule apply to functions inside a logarithm?
4. Can we extend this to find the slope in the $y$-direction?
5. How do partial derivatives relate to the gradient of a surface?

Tip: For multi-variable functions, partial derivatives help you analyze how the function changes with respect to one variable while keeping others constant.