1. **State the problem:** Find the derivative $f'(x)$ of the function $$y = \ln \left( \frac{(x+1)^4 (x^3+2)}{x-1} \right)$$ using logarithmic differentiation. 2. **Recall the logarithmic differentiation formula:** For $y = \ln u$, the derivative is $$y' = \frac{u'}{u}$$. 3. **Apply logarithm properties:** Use the property $$\ln \left( \frac{A}{B} \right) = \ln A - \ln B$$ and $$\ln (AB) = \ln A + \ln B$$ to rewrite: $$y = \ln (x+1)^4 + \ln (x^3+2) - \ln (x-1)$$ 4. **Simplify using logarithm power rule:** $$\ln (x+1)^4 = 4 \ln (x+1)$$, so $$y = 4 \ln (x+1) + \ln (x^3+2) - \ln (x-1)$$ 5. **Differentiate term-by-term:** $$y' = 4 \cdot \frac{1}{x+1} + \frac{3x^2}{x^3+2} - \frac{1}{x-1}$$ 6. **Combine into a single expression:** $$f'(x) = \frac{4}{x+1} + \frac{3x^2}{x^3+2} - \frac{1}{x-1}$$ This is the derivative of the given function using logarithmic differentiation.