1. **State the problem:** Find the derivative of the function $$y = \frac{8t - 7}{5t + 1}$$ using the quotient rule. 2. **Recall the quotient rule formula:** If $$y = \frac{u}{v}$$, then $$\frac{dy}{dt} = \frac{v \frac{du}{dt} - u \frac{dv}{dt}}{v^2}$$. 3. **Identify $$u$$ and $$v$$:** Here, $$u = 8t - 7$$ and $$v = 5t + 1$$. 4. **Compute derivatives:** $$\frac{du}{dt} = 8$$ and $$\frac{dv}{dt} = 5$$. 5. **Apply the quotient rule:** $$\frac{dy}{dt} = \frac{(5t + 1)(8) - (8t - 7)(5)}{(5t + 1)^2}$$ 6. **Expand the numerator:** $$= \frac{40t + 8 - (40t - 35)}{(5t + 1)^2}$$ 7. **Simplify the numerator:** $$= \frac{40t + 8 - 40t + 35}{(5t + 1)^2} = \frac{43}{(5t + 1)^2}$$ 8. **Final answer:** $$\frac{dy}{dt} = \frac{43}{(5t + 1)^2}$$ This matches the given solution, confirming the derivative is correct.