Найти производную. F (x) = (х^2+x+1) / (x-1) ^2

Найдем производную, воспользовавшись правилом нахождения производной произведения функций.

f' (x) = [ (х² + x + 1) / (x - 1) ²]' = (х² + x + 1) ' * (x - 1) ⁻² + (х² + x + 1) * [ (x - 1) ⁻²]' =

= [ (х²) ' + x' + (1) '] * (x - 1) ⁻² + (х² + x + 1) * [-2 * (x - 1) ⁻²⁻¹ * (x - 1) '] =

= (2x + 1 + 0) * (x - 1) ⁻² + (х² + x + 1) * [-2 * (x - 1) ⁻³ * 1] =

= (2x + 1) / (x - 1) ² - 2 * (х² + x + 1) / (x - 1) ³ =

= (2x + 1) * (x - 1) / (x - 1) ³ - (2 х² + 2x + 2) / (x - 1) ³ =

= [ (2x² + x - 2x - 1) - (2 х² + 2x + 2) ] / (x - 1) ³ =

= (2x² - x - 1 - 2x² - 2x - 2) / (x - 1) ³ = (-3x - 3) / (x - 1) ³ = 3 * (x + 1) / (x - 1) ³.