1. **State the problem:** Given the equation $$xy + 7x - 8x^2 = 5$$, we need to find the derivative $y'$ by implicit differentiation and then solve for $y$ explicitly and differentiate to find $y'$ again. 2. **Implicit differentiation:** Differentiate both sides with respect to $x$. Use the product rule for $xy$: $$\frac{d}{dx}(xy) + \frac{d}{dx}(7x) - \frac{d}{dx}(8x^2) = \frac{d}{dx}(5)$$ 3. Applying the product rule to $xy$: $$x \frac{dy}{dx} + y \frac{dx}{dx} + 7 - 16x = 0$$ which is $$x y' + y + 7 - 16x = 0$$ 4. Solve for $y'$: $$x y' = 16x - 7 - y$$ $$y' = \frac{16x - 7 - y}{x}$$ 5. **Solve for $y$ explicitly:** $$xy + 7x - 8x^2 = 5$$ $$xy = 5 - 7x + 8x^2$$ $$y = \frac{5 - 7x + 8x^2}{x}$$ 6. Simplify $y$: $$y = \frac{5}{x} - 7 + 8x$$ 7. **Differentiate $y$ explicitly:** $$y' = \frac{d}{dx}\left(\frac{5}{x} - 7 + 8x\right) = -\frac{5}{x^2} + 0 + 8 = 8 - \frac{5}{x^2}$$ 8. **Check consistency:** Substitute $y$ from step 6 into the implicit derivative formula from step 4: $$y' = \frac{16x - 7 - \left(\frac{5}{x} - 7 + 8x\right)}{x} = \frac{16x - 7 - \frac{5}{x} + 7 - 8x}{x} = \frac{8x - \frac{5}{x}}{x} = \frac{8x^2 - 5}{x^2} = 8 - \frac{5}{x^2}$$ This matches the explicit derivative. **Final answers:** (a) $$y' = \frac{16x - 7 - y}{x}$$ (b) $$y' = 8 - \frac{5}{x^2}$$