1. **State the problem:** Differentiate the function $$y = \frac{(x+1)(x-2)^3}{x-3}$$ using logarithmic differentiation. 2. **Recall the formula and rules:** Logarithmic differentiation involves taking the natural logarithm of both sides, then differentiating implicitly. This is useful for functions that are products, quotients, or powers. 3. **Take the natural logarithm of both sides:** $$\ln y = \ln \left( \frac{(x+1)(x-2)^3}{x-3} \right)$$ Using log properties: $$\ln y = \ln(x+1) + \ln((x-2)^3) - \ln(x-3)$$ $$\ln y = \ln(x+1) + 3\ln(x-2) - \ln(x-3)$$ 4. **Differentiate both sides with respect to $x$:** $$\frac{1}{y} \frac{dy}{dx} = \frac{1}{x+1} + 3 \cdot \frac{1}{x-2} - \frac{1}{x-3}$$ 5. **Solve for $\frac{dy}{dx}$:** $$\frac{dy}{dx} = y \left( \frac{1}{x+1} + \frac{3}{x-2} - \frac{1}{x-3} \right)$$ 6. **Substitute back $y$:** $$\frac{dy}{dx} = \frac{(x+1)(x-2)^3}{x-3} \left( \frac{1}{x+1} + \frac{3}{x-2} - \frac{1}{x-3} \right)$$ 7. **Simplify the expression inside the parentheses:** $$\frac{1}{x+1} + \frac{3}{x-2} - \frac{1}{x-3}$$ This is the derivative in simplified logarithmic differentiation form. **Final answer:** $$\boxed{\frac{dy}{dx} = \frac{(x+1)(x-2)^3}{x-3} \left( \frac{1}{x+1} + \frac{3}{x-2} - \frac{1}{x-3} \right)}$$