1. **State the problem:** We are given the derivative $ \frac{dy}{dx} = \frac{2x - 6}{x^2 - 2x} $ and asked to find the original function $ y $. 2. **Recall the formula:** To find $ y $, we need to integrate the derivative: $$ y = \int \frac{2x - 6}{x^2 - 2x} \, dx + C $$ where $ C $ is the constant of integration. 3. **Simplify the integrand:** Factor the denominator: $$ x^2 - 2x = x(x - 2) $$ Rewrite the integrand: $$ \frac{2x - 6}{x(x - 2)} $$ 4. **Use partial fraction decomposition:** Assume $$ \frac{2x - 6}{x(x - 2)} = \frac{A}{x} + \frac{B}{x - 2} $$ Multiply both sides by $ x(x - 2) $: $$ 2x - 6 = A(x - 2) + Bx $$ 5. **Find constants $ A $ and $ B $:** Set $ x = 0 $: $$ 2(0) - 6 = A(0 - 2) + B(0) \Rightarrow -6 = -2A \Rightarrow A = 3 $$ Set $ x = 2 $: $$ 2(2) - 6 = A(2 - 2) + B(2) \Rightarrow 4 - 6 = 0 + 2B \Rightarrow -2 = 2B \Rightarrow B = -1 $$ 6. **Rewrite the integral:** $$ y = \int \left( \frac{3}{x} - \frac{1}{x - 2} \right) dx + C $$ 7. **Integrate term-by-term:** $$ y = 3 \int \frac{1}{x} dx - \int \frac{1}{x - 2} dx + C $$ $$ y = 3 \ln|x| - \ln|x - 2| + C $$ 8. **Final answer:** $$ \boxed{y = 3 \ln|x| - \ln|x - 2| + C} $$ This is the original function $ y $ whose derivative is given.