1. **State the problem:** Differentiate the function $$y = (At + B)e^{-t} + 4e^{\frac{1}{2}t}$$ with respect to $$t$$. 2. **Recall the differentiation rules:** - Product rule: $$\frac{d}{dt}[u(t)v(t)] = u'(t)v(t) + u(t)v'(t)$$ - Exponential rule: $$\frac{d}{dt}[e^{kt}] = ke^{kt}$$ 3. **Differentiate the first term:** Let $$u = At + B$$ and $$v = e^{-t}$$. Calculate $$u' = A$$ and $$v' = -e^{-t}$$. Using product rule: $$\frac{d}{dt}[(At + B)e^{-t}] = A e^{-t} + (At + B)(-e^{-t}) = A e^{-t} - (At + B)e^{-t}$$ 4. **Simplify the first term's derivative:** $$A e^{-t} - (At + B)e^{-t} = e^{-t}(A - At - B) = e^{-t}(-At + A - B)$$ 5. **Differentiate the second term:** $$\frac{d}{dt}[4e^{\frac{1}{2}t}] = 4 \cdot \frac{1}{2} e^{\frac{1}{2}t} = 2 e^{\frac{1}{2}t}$$ 6. **Combine the derivatives:** $$\frac{dy}{dt} = e^{-t}(-At + A - B) + 2 e^{\frac{1}{2}t}$$ **Final answer:** $$\boxed{\frac{dy}{dt} = e^{-t}(-At + A - B) + 2 e^{\frac{1}{2}t}}$$