1. **State the problem:** We have a function $$Z = e^x \sin y$$ where $$x = u v^2$$ and $$y = v u^2$$. We need to find the partial derivatives $$\frac{\partial Z}{\partial u}$$ and $$\frac{\partial Z}{\partial v}$$ at $$u=1$$ and $$v=\frac{\pi}{2}$$. 2. **Recall the chain rule for partial derivatives:** Since $$Z$$ depends on $$x$$ and $$y$$, which in turn depend on $$u$$ and $$v$$, we use: $$\frac{\partial Z}{\partial u} = \frac{\partial Z}{\partial x} \frac{\partial x}{\partial u} + \frac{\partial Z}{\partial y} \frac{\partial y}{\partial u}$$ $$\frac{\partial Z}{\partial v} = \frac{\partial Z}{\partial x} \frac{\partial x}{\partial v} + \frac{\partial Z}{\partial y} \frac{\partial y}{\partial v}$$ 3. **Calculate partial derivatives of $$Z$$ with respect to $$x$$ and $$y$$:** $$\frac{\partial Z}{\partial x} = e^x \sin y$$ $$\frac{\partial Z}{\partial y} = e^x \cos y$$ 4. **Calculate partial derivatives of $$x$$ and $$y$$ with respect to $$u$$ and $$v$$:** $$x = u v^2 \Rightarrow \frac{\partial x}{\partial u} = v^2, \quad \frac{\partial x}{\partial v} = 2 u v$$ $$y = v u^2 \Rightarrow \frac{\partial y}{\partial u} = 2 u v, \quad \frac{\partial y}{\partial v} = u^2$$ 5. **Evaluate all derivatives at $$u=1$$ and $$v=\frac{\pi}{2}$$:** $$x = 1 \times \left(\frac{\pi}{2}\right)^2 = \frac{\pi^2}{4}$$ $$y = \frac{\pi}{2} \times 1^2 = \frac{\pi}{2}$$ $$\frac{\partial x}{\partial u} = \left(\frac{\pi}{2}\right)^2 = \frac{\pi^2}{4}$$ $$\frac{\partial x}{\partial v} = 2 \times 1 \times \frac{\pi}{2} = \pi$$ $$\frac{\partial y}{\partial u} = 2 \times 1 \times \frac{\pi}{2} = \pi$$ $$\frac{\partial y}{\partial v} = 1^2 = 1$$ 6. **Calculate $$\frac{\partial Z}{\partial x}$$ and $$\frac{\partial Z}{\partial y}$$ at these values:** $$\frac{\partial Z}{\partial x} = e^{\frac{\pi^2}{4}} \sin \left(\frac{\pi}{2}\right) = e^{\frac{\pi^2}{4}} \times 1 = e^{\frac{\pi^2}{4}}$$ $$\frac{\partial Z}{\partial y} = e^{\frac{\pi^2}{4}} \cos \left(\frac{\pi}{2}\right) = e^{\frac{\pi^2}{4}} \times 0 = 0$$ 7. **Compute $$\frac{\partial Z}{\partial u}$$:** $$\frac{\partial Z}{\partial u} = e^{\frac{\pi^2}{4}} \times \frac{\pi^2}{4} + 0 \times \pi = \frac{\pi^2}{4} e^{\frac{\pi^2}{4}}$$ 8. **Compute $$\frac{\partial Z}{\partial v}$$:** $$\frac{\partial Z}{\partial v} = e^{\frac{\pi^2}{4}} \times \pi + 0 \times 1 = \pi e^{\frac{\pi^2}{4}}$$ **Final answers:** $$\boxed{\frac{\partial Z}{\partial u} = \frac{\pi^2}{4} e^{\frac{\pi^2}{4}}, \quad \frac{\partial Z}{\partial v} = \pi e^{\frac{\pi^2}{4}}}$$