1. **State the problem:** Differentiate the function $$f(t) = (5t + e^t)(3 - \sqrt{t})$$ with respect to $$t$$. 2. **Formula used:** Use the product rule for differentiation: $$\frac{d}{dt}[u(t)v(t)] = u'(t)v(t) + u(t)v'(t)$$ where $$u(t) = 5t + e^t$$ and $$v(t) = 3 - \sqrt{t}$$. 3. **Find derivatives of each part:** - $$u'(t) = \frac{d}{dt}(5t) + \frac{d}{dt}(e^t) = 5 + e^t$$ - $$v'(t) = \frac{d}{dt}(3) - \frac{d}{dt}(t^{1/2}) = 0 - \frac{1}{2}t^{-1/2} = -\frac{1}{2\sqrt{t}}$$ 4. **Apply the product rule:** $$f'(t) = (5 + e^t)(3 - \sqrt{t}) + (5t + e^t)\left(-\frac{1}{2\sqrt{t}}\right)$$ 5. **Simplify the expression:** $$f'(t) = (5 + e^t)(3 - \sqrt{t}) - \frac{5t + e^t}{2\sqrt{t}}$$ 6. **Final answer:** $$\boxed{f'(t) = (5 + e^t)(3 - \sqrt{t}) - \frac{5t + e^t}{2\sqrt{t}}}$$