1. **State the problem:** Differentiate the function $$y = \frac{x+1}{x^2 - 1}$$ with respect to $$x$$. 2. **Recall the quotient rule:** For a function $$y = \frac{u}{v}$$, the derivative is $$y' = \frac{v u' - u v'}{v^2}$$. 3. **Identify $$u$$ and $$v$$:** $$u = x+1$$ $$v = x^2 - 1$$ 4. **Compute derivatives:** $$u' = \frac{d}{dx}(x+1) = 1$$ $$v' = \frac{d}{dx}(x^2 - 1) = 2x$$ 5. **Apply quotient rule:** $$y' = \frac{(x^2 - 1)(1) - (x+1)(2x)}{(x^2 - 1)^2}$$ 6. **Simplify numerator:** $$= \frac{x^2 - 1 - 2x(x+1)}{(x^2 - 1)^2}$$ $$= \frac{x^2 - 1 - 2x^2 - 2x}{(x^2 - 1)^2}$$ 7. **Combine like terms:** $$= \frac{x^2 - 2x^2 - 2x - 1}{(x^2 - 1)^2} = \frac{-x^2 - 2x - 1}{(x^2 - 1)^2}$$ 8. **Factor numerator:** $$-x^2 - 2x - 1 = -(x^2 + 2x + 1) = -(x+1)^2$$ 9. **Final derivative:** $$y' = \frac{-(x+1)^2}{(x^2 - 1)^2}$$