1. **State the problem:** Find the first derivative of the function $$y = e^x \arctan(e^x)$$. 2. **Formula used:** We will use the product rule for derivatives, which states: $$\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)$$ where $$u(x) = e^x$$ and $$v(x) = \arctan(e^x)$$. 3. **Find derivatives of each part:** - Derivative of $$u(x) = e^x$$ is $$u'(x) = e^x$$. - To find $$v'(x)$$, use the chain rule: $$v'(x) = \frac{1}{1+(e^x)^2} \cdot \frac{d}{dx}(e^x) = \frac{1}{1+e^{2x}} \cdot e^x = \frac{e^x}{1+e^{2x}}$$. 4. **Apply the product rule:** $$y' = u'(x)v(x) + u(x)v'(x) = e^x \arctan(e^x) + e^x \cdot \frac{e^x}{1+e^{2x}}$$ 5. **Simplify the expression:** $$y' = e^x \arctan(e^x) + \frac{e^{2x}}{1+e^{2x}}$$ **Final answer:** $$\boxed{y' = e^x \arctan(e^x) + \frac{e^{2x}}{1+e^{2x}}}$$