1. **State the problem:** Find the slope of the tangent line to the graph of $f(x) = x e^x$ at the point $(1, e)$. 2. **Recall the formula:** The slope of the tangent line at a point is the derivative of the function evaluated at that point. 3. **Find the derivative:** Use the product rule for derivatives: if $f(x) = u(x)v(x)$, then $$f'(x) = u'(x)v(x) + u(x)v'(x).$$ Here, $u(x) = x$ and $v(x) = e^x$. 4. Compute derivatives: $$u'(x) = 1,$$ $$v'(x) = e^x.$$ 5. Apply the product rule: $$f'(x) = 1 \cdot e^x + x \cdot e^x = e^x + x e^x = e^x(1 + x).$$ 6. **Evaluate the derivative at $x=1$:** $$f'(1) = e^1 (1 + 1) = e \times 2 = 2e.$$ 7. **Conclusion:** The slope of the tangent line to the graph at the point $(1, e)$ is $2e$.