1. We are asked to find the derivative of the function $$y = (2x - 5)^{-1} (x^2 - 5x)^6$$. 2. Notice this is a product of two functions: $$u = (2x - 5)^{-1}$$ and $$v = (x^2 - 5x)^6$$. 3. Use the product rule: $$y' = u'v + uv'$$. 4. Differentiate $$u$$: $$u = (2x - 5)^{-1}$$ Using the chain rule: $$u' = -1 (2x - 5)^{-2} imes 2 = -2(2x-5)^{-2}$$. 5. Differentiate $$v$$: $$v = (x^2 - 5x)^6$$. Using the chain rule: $$v' = 6(x^2 - 5x)^5 imes (2x - 5)$$. 6. Substitute into the product rule formula: $$y' = -2(2x-5)^{-2} (x^2 - 5x)^6 + (2x-5)^{-1} imes 6(x^2 - 5x)^5 (2x - 5)$$. 7. Simplify the second term: $$(2x-5)^{-1} imes (2x - 5) = 1$$ So it becomes: $$6 (x^2 - 5x)^5$$. 8. Final derivative: $$y' = -2(2x-5)^{-2} (x^2 - 5x)^6 + 6 (x^2 - 5x)^5$$. 9. We can factor out $$ (x^2 - 5x)^5 $$: $$y' = (x^2 - 5x)^5 \left(-2(2x-5)^{-2} (x^2 - 5x) + 6 \right)$$. This is the derivative of the given function.