1. **State the problem:** Find the first partial derivative of the function $$f(x,y) = y^2 e^x + \tan(xy)$$ with respect to $$x$$. 2. **Recall the formula and rules:** - The partial derivative with respect to $$x$$ treats $$y$$ as a constant. - Derivative of $$e^x$$ with respect to $$x$$ is $$e^x$$. - Derivative of $$\tan(u)$$ with respect to $$x$$ is $$\sec^2(u) \cdot \frac{du}{dx}$$ by chain rule. 3. **Apply the derivative:** - For the first term $$y^2 e^x$$, since $$y^2$$ is constant with respect to $$x$$, derivative is $$y^2 e^x$$. - For the second term $$\tan(xy)$$, let $$u = xy$$. Then $$\frac{du}{dx} = y$$. So derivative is $$\sec^2(xy) \cdot y$$. 4. **Combine results:** $$\frac{\partial f}{\partial x} = y^2 e^x + y \sec^2(xy)$$ 5. **Explanation:** - We treat $$y$$ as a constant when differentiating with respect to $$x$$. - The chain rule is used for the $$\tan(xy)$$ term. **Final answer:** $$\boxed{\frac{\partial f}{\partial x} = y^2 e^x + y \sec^2(xy)}$$