1. **State the problem:** Find the first partial derivative of the function $$f(x,y) = e^{x+y} + y^2 \sin(x)$$ with respect to $$x$$. 2. **Recall the formula and rules:** - The partial derivative with respect to $$x$$ treats $$y$$ as a constant. - Derivative of $$e^{x+y}$$ with respect to $$x$$ is $$e^{x+y}$$ because $$y$$ is constant. - Derivative of $$y^2 \sin(x)$$ with respect to $$x$$ is $$y^2 \cos(x)$$ since $$y^2$$ is constant and derivative of $$\sin(x)$$ is $$\cos(x)$$. 3. **Calculate the partial derivative:** $$\frac{\partial f}{\partial x} = \frac{\partial}{\partial x} \left(e^{x+y} + y^2 \sin(x)\right) = \frac{\partial}{\partial x} e^{x+y} + \frac{\partial}{\partial x} \left(y^2 \sin(x)\right)$$ 4. **Apply derivatives:** $$= e^{x+y} + y^2 \cos(x)$$ 5. **Final answer:** $$\boxed{\frac{\partial f}{\partial x} = e^{x+y} + y^2 \cos(x)}$$