Rationale Funktion/Ableitung/1/Aufgabe/Lösung

Nach der Quotientenregel ist

{}{\begin{aligned}f'(x)&={\frac {{\left(x^{3}-9x^{2}-x+5\right)}'{\left(x^{2}-6x+5\right)}-{\left(x^{3}-9x^{2}-x+5\right)}{\left(x^{2}-6x+5\right)}'}{{\left(x^{2}-6x+5\right)}^{2}}}\\&={\frac {{\left(3x^{2}-18x-1\right)}{\left(x^{2}-6x+5\right)}-{\left(x^{3}-9x^{2}-x+5\right)}{\left(2x-6\right)}}{x^{4}-12x^{3}+46x^{2}-60x+25}}\\&={\frac {3x^{4}-36x^{3}-38x^{2}-96x-5-{\left(2x^{4}-24x^{3}+52x^{2}+16x-30\right)}}{x^{4}-12x^{3}+90x^{2}-112x+25}}\\&={\frac {x^{4}-12x^{3}-38x^{2}-96x+25}{x^{4}-12x^{3}+46x^{2}-60x+25}}.\end{aligned}}

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