The given problem involves finding the partial derivative of the function $f = e^{-xy}$ with respect to $y$.

Solution:

The function is: $f = e^{-xy}$

To find $\frac{\partial f}{\partial y}$, we use the chain rule. The derivative of $e^u$ with respect to $u$ is $e^u$, and the chain rule applies for the exponent $u = -xy$:

1. Differentiate $e^{-xy}$: $\frac{\partial f}{\partial y} = e^{-xy} \cdot \frac{\partial (-xy)}{\partial y}.$

2. Compute $\frac{\partial (-xy)}{\partial y}$: $\frac{\partial (-xy)}{\partial y} = -x.$

3. Substitute this back: $\frac{\partial f}{\partial y} = e^{-xy} \cdot (-x).$

4. Simplify: $\frac{\partial f}{\partial y} = -x e^{-xy}.$

Final Answer:

$\frac{\partial f}{\partial y} = -x e^{-xy}.$

Let me know if you want further clarifications or have additional questions.

Follow-Up Questions:

1. How is the chain rule applied in partial differentiation?
2. What happens if $x = 0$ in this function?
3. Can you find $\frac{\partial^2 f}{\partial y^2}$?
4. How would $\frac{\partial f}{\partial x}$ differ from $\frac{\partial f}{\partial y}$?
5. How does the exponential function $e^{-xy}$ behave as $x$ or $y$ increases?

Tip:

When taking partial derivatives, treat all other variables as constants except the one you are differentiating with respect to.