1. **State the problem:** We need to find the derivative $\frac{dz}{dx}$ when $z = x e^{xy}$, where $y$ is treated as a constant with respect to $x$. 2. **Recall the formula:** To differentiate a product of two functions, use the product rule: $$\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)$$ 3. **Identify functions:** Here, $u(x) = x$ and $v(x) = e^{xy}$. 4. **Differentiate each part:** - $u'(x) = 1$ - To find $v'(x)$, use the chain rule: $$v'(x) = \frac{d}{dx} e^{xy} = e^{xy} \cdot \frac{d}{dx}(xy)$$ Since $y$ is constant, $$\frac{d}{dx}(xy) = y$$ So, $$v'(x) = y e^{xy}$$ 5. **Apply product rule:** $$\frac{dz}{dx} = u'(x)v(x) + u(x)v'(x) = 1 \cdot e^{xy} + x \cdot y e^{xy} = e^{xy} + x y e^{xy}$$ 6. **Factor common terms:** $$\frac{dz}{dx} = e^{xy}(1 + xy)$$ **Final answer:** $$\boxed{\frac{dz}{dx} = e^{xy}(1 + xy)}$$