\:Let\:z\:=\:f\left(x,y\right)\:=\:exy;\:x\:=\:2u+v\:and\:y\:=\:u\:Find\:the\:slope\:of\:the\:surface\:f\left(x,y\right)\:in\:the\:y-direction\:at\:the\:point\:\left(3,0\right).

To find the slope of the surface \( z = f(x,y) = e^{xy} \) in the \( y \)-direction at the point \( (3,0) \), we need to compute the partial derivative of \( f(x,y) \) with respect to \( y \) and then evaluate it at the given point.

### Step 1: Compute the partial derivative of \( f(x,y) \) with respect to \( y \).

Given:
\[
z = f(x,y) = e^{xy}
\]
To find \( \frac{\partial z}{\partial y} \), we differentiate \( z \) with respect to \( y \):

\[
\frac{\partial z}{\partial y} = \frac{\partial}{\partial y} \left( e^{xy} \right)
\]

Using the chain rule:

\[
\frac{\partial z}{\partial y} = e^{xy} \cdot \frac{\partial}{\partial y} (xy)
\]

Since \( \frac{\partial}{\partial y} (xy) = x \), we have:

\[
\frac{\partial z}{\partial y} = e^{xy} \cdot x
\]

### Step 2: Evaluate the partial derivative at the given point.

We need to evaluate \( \frac{\partial z}{\partial y} \) at the point \( (x,y) = (3,0) \):

\[
\frac{\partial z}{\partial y}\Big|_{(x,y) = (3,0)} = e^{3 \cdot 0} \cdot 3 = e^0 \cdot 3 = 1 \cdot 3 = 3
\]

### Final Answer:
The slope of the surface \( f(x,y) \) in the \( y \)-direction at the point \( (3,0) \) is \( \boxed{3} \).

Would you like more details on any of the steps? Here are some additional related questions you might find interesting:

1. How do you find the partial derivative of a function with multiple variables?
2. What is the geometric interpretation of a partial derivative?
3. How do you find the partial derivative with respect to \( x \) at a specific point?
4. How do you determine the slope in a different direction (e.g., in the \( x \)-direction)?
5. What does the partial derivative tell us about the behavior of the function near the given point?

**Tip:** Always remember that partial derivatives give the rate of change of the function with respect to one variable while keeping the other variables constant.

Find the slope of the surface f(x,y) = e^(xy) in the y-direction at the point (3,0).

Explore how to find the slope of the surface z = e^(xy) in the y-direction at the point (3,0) using partial derivatives. This solution demonstrates key calculus concepts, including the chain rule and partial differentiation. Suitable for undergraduate-level students.

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