1. **State the problem:** Find the differential of the function $$y=\frac{3x - 7}{5x + 4}$$. 2. **Recall the formula:** For a function $$y=\frac{u}{v}$$, the derivative is given by the quotient rule: $$\frac{dy}{dx} = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}$$ where $$u=3x - 7$$ and $$v=5x + 4$$. 3. **Calculate derivatives of numerator and denominator:** $$\frac{du}{dx} = 3$$ $$\frac{dv}{dx} = 5$$ 4. **Apply the quotient rule:** $$\frac{dy}{dx} = \frac{(5x + 4)(3) - (3x - 7)(5)}{(5x + 4)^2}$$ 5. **Expand the numerator:** $$= \frac{15x + 12 - (15x - 35)}{(5x + 4)^2}$$ 6. **Simplify the numerator:** $$= \frac{15x + 12 - 15x + 35}{(5x + 4)^2} = \frac{47}{(5x + 4)^2}$$ 7. **Final answer:** $$\boxed{\frac{dy}{dx} = \frac{47}{(5x + 4)^2}}$$