Ableitungsrechner

Übungen zur Quotientenregel

Hier kannst du die Anwendung der Quotientenregel üben. Um Zähler und Nenner abzuleiten, musst du oft weitere Regeln wie die Ketten- oder Produktregel anwenden.

✏️ Leichte Aufgaben

Fokus auf der grundlegenden Anwendung der Quotientenregel mit einfachen Brüchen.

  1. $f(x) = \frac{x^{2}}{x + 1}$ Rechner
    $f'(x) = \frac{2\,x\left(x + 1\right) - x^{2}}{\left(x + 1\right)^{2}}$
  2. $f(x) = \frac{x + 1}{x - 1}$ Rechner
    $f'(x) = -\frac{2}{\left(x - 1\right)^{2}}$
  3. $f(x) = \frac{x}{\mathrm{e}^{x}}$ Rechner
    $f'(x) = \frac{1 - x}{\mathrm{e}^{x}}$
  4. $f(x) = \frac{\sin\left(x\right)}{\cos\left(x\right)}$ Rechner
    $f'(x) = \frac{1}{\cos^{2}\left(x\right)}$
  5. $f(x) = \frac{\ln\left(x\right)}{x}$ Rechner
    $f'(x) = \frac{1 - \ln\left(x\right)}{x^{2}}$
  6. $f(x) = \frac{2\,x + 3}{4\,x - 1}$ Rechner
    $f'(x) = -\frac{14}{\left(4\,x - 1\right)^{2}}$
  7. $f(x) = \frac{x^{3}}{x^{2} + 1}$ Rechner
    $f'(x) = \frac{x^{4} + 3\,x^{2}}{\left(x^{2} + 1\right)^{2}}$
  8. $f(x) = \frac{\mathrm{e}^{x}}{x^{2}}$ Rechner
    $f'(x) = \frac{x\mathrm{e}^{x} - 2\,\mathrm{e}^{x}}{x^{3}}$
  9. $f(x) = \frac{5}{x^{3}}$ Rechner
    $f'(x) = -\frac{15}{x^{4}}$
  10. $f(x) = \frac{\cos\left(x\right)}{x}$ Rechner
    $f'(x) = \frac{-x\sin\left(x\right) - \cos\left(x\right)}{x^{2}}$
  11. $f(x) = \frac{x^{2} - 3\,x + 2}{x + 4}$ Rechner
    $f'(x) = \frac{x^{2} + 8\,x - 14}{\left(x + 4\right)^{2}}$
  12. $f(x) = \frac{x}{\ln\left(x\right)}$ Rechner
    $f'(x) = \frac{\ln\left(x\right) - 1}{\left(\ln\left(x\right)\right)^{2}}$
  13. $f(x) = \frac{\mathrm{e}^{x}}{\mathrm{e}^{x} + 1}$ Rechner
    $f'(x) = \frac{\mathrm{e}^{x}}{\left(\mathrm{e}^{x} + 1\right)^{2}}$
  14. $f(x) = \frac{x^{4} + 1}{x^{2}}$ Rechner
    $f'(x) = 2\,x - \frac{2}{x^{3}}$
  15. $f(x) = \frac{\sin\left(x\right)}{\mathrm{e}^{x}}$ Rechner
    $f'(x) = \frac{\cos\left(x\right) - \sin\left(x\right)}{\mathrm{e}^{x}}$

🔥 Mittelschwere Aufgaben

Kombination der Quotientenregel mit verschiedenen Funktionstypen im Zähler und Nenner.

  1. $f(x) = \frac{\sqrt{x}}{\sin\left(x\right)}$ Rechner
    $f'(x) = \frac{\frac{\sin\left(x\right)}{2\,\sqrt{x}} - \sqrt{x}\cos\left(x\right)}{\sin^{2}\left(x\right)}$
  2. $f(x) = \frac{\ln\left(x\right)}{x^{3}}$ Rechner
    $f'(x) = \frac{1 - 3\,\ln\left(x\right)}{x^{4}}$
  3. $f(x) = \frac{x^{2}}{\mathrm{e}^{-x}}$ Rechner
    $f'(x) = \mathrm{e}^{x}\left(2\,x + x^{2}\right)$
  4. $f(x) = \frac{\cos\left(2\,x\right)}{\mathrm{e}^{x}}$ Rechner
    $f'(x) = \frac{-2\,\sin\left(2\,x\right) - \cos\left(2\,x\right)}{\mathrm{e}^{x}}$
  5. $f(x) = \frac{\tan\left(x\right)}{x}$ Rechner
    $f'(x) = \frac{\frac{x}{\cos^{2}\left(x\right)} - \tan\left(x\right)}{x^{2}}$
  6. $f(x) = \frac{x^{2}}{2\,\ln\left(x\right)}$ Rechner
    $f'(x) = \frac{2\,x\ln\left(x\right) - x}{2\,\left(\ln\left(x\right)\right)^{2}}$
  7. $f(x) = \frac{x + 1}{\sqrt{x}}$ Rechner
    $f'(x) = \frac{x - 1}{2\,x\sqrt{x}}$
  8. $f(x) = \frac{x}{\arcsin\left(x\right)}$ Rechner
    $f'(x) = \frac{\arcsin\left(x\right) - \frac{x}{\sqrt{1 - x^{2}}}}{\left(\arcsin\left(x\right)\right)^{2}}$
  9. $f(x) = \frac{\mathrm{e}^{2\,x}}{x}$ Rechner
    $f'(x) = \frac{2\,x\mathrm{e}^{2\,x} - \mathrm{e}^{2\,x}}{x^{2}}$
  10. $f(x) = \frac{\ln\left(x\right)}{\cos\left(x\right)}$ Rechner
    $f'(x) = \frac{\frac{\cos\left(x\right)}{x} + \ln\left(x\right)\sin\left(x\right)}{\cos^{2}\left(x\right)}$
  11. $f(x) = \frac{x^{3}}{2^{x}}$ Rechner
    $f'(x) = \frac{3\,x^{2} - x^{3}\ln\left(2\right)}{2^{x}}$
  12. $f(x) = \frac{\sin\left(x\right)}{\sin\left(x\right) + \cos\left(x\right)}$ Rechner
    $f'(x) = \frac{1}{\left(\sin\left(x\right) + \cos\left(x\right)\right)^{2}}$
  13. $f(x) = \frac{x}{\arctan\left(x\right)}$ Rechner
    $f'(x) = \frac{\arctan\left(x\right) - \frac{x}{1 + x^{2}}}{\left(\arctan\left(x\right)\right)^{2}}$
  14. $f(x) = \frac{x^{2} + 1}{\sqrt[3]{x}}$ Rechner
    $f'(x) = \frac{5\,x^{2} - 1}{3\,x^{\frac{4}{3}}}$
  15. $f(x) = \frac{\log_{10}{\left(x\right)}}{x}$ Rechner
    $f'(x) = \frac{\frac{1}{\ln\left(10\right)} - \log_{10}{\left(x\right)}}{x^{2}}$

🚀 Schwere Aufgaben

Hier muss die Quotientenregel auf Brüche angewendet werden, deren Zähler oder Nenner selbst komplexe Ableitungen erfordern.

  1. $f(x) = \frac{\sin\left(x^{2}\right)}{\mathrm{e}^{2\,x}}$ Rechner
    $f'(x) = \frac{2\,x\cos\left(x^{2}\right) - 2\,\sin\left(x^{2}\right)}{\mathrm{e}^{2\,x}}$
  2. $f(x) = \frac{\sqrt{x}}{\ln\left(x^{2} + 1\right)}$ Rechner
    $f'(x) = \frac{\frac{\ln\left(x^{2} + 1\right)}{2\,\sqrt{x}} - \frac{2\,x\sqrt{x}}{x^{2} + 1}}{\left(\ln\left(x^{2} + 1\right)\right)^{2}}$
  3. $f(x) = \frac{x^{3}}{\sin\left(\ln\left(x\right)\right)}$ Rechner
    $f'(x) = \frac{3\,x^{2}\sin\left(\ln\left(x\right)\right) - x^{2}\cos\left(\ln\left(x\right)\right)}{\sin^{2}\left(\ln\left(x\right)\right)}$
  4. $f(x) = \frac{\cos\left(3\,x\right)}{\arctan\left(2\,x\right)}$ Rechner
    $f'(x) = \frac{-3\,\sin\left(3\,x\right)\arctan\left(2\,x\right) - \frac{2\,\cos\left(3\,x\right)}{1 + 4\,x^{2}}}{\left(\arctan\left(2\,x\right)\right)^{2}}$
  5. $f(x) = \frac{x\mathrm{e}^{x}}{\sin\left(x\right)}$ Rechner
    $f'(x) = \frac{\mathrm{e}^{x}\left(1 + x\right)\sin\left(x\right) - x\mathrm{e}^{x}\cos\left(x\right)}{\sin^{2}\left(x\right)}$
  6. $f(x) = \frac{\ln\left(x\right)}{\sqrt{x^{2} + 1}}$ Rechner
    $f'(x) = \frac{\frac{\sqrt{x^{2} + 1}}{x} - \frac{x\ln\left(x\right)}{\sqrt{x^{2} + 1}}}{x^{2} + 1}$
  7. $f(x) = \frac{x}{\cos\left(\mathrm{e}^{x}\right)}$ Rechner
    $f'(x) = \frac{\cos\left(\mathrm{e}^{x}\right) + x\mathrm{e}^{x}\sin\left(\mathrm{e}^{x}\right)}{\cos^{2}\left(\mathrm{e}^{x}\right)}$
  8. $f(x) = \frac{\sin\left(x\right)}{\left(x + 1\right)^{2}}$ Rechner
    $f'(x) = \frac{\cos\left(x\right)\left(x + 1\right) - 2\,\sin\left(x\right)}{\left(x + 1\right)^{3}}$
  9. $f(x) = \frac{\mathrm{e}^{x}}{\arcsin\left(\sqrt{x}\right)}$ Rechner
    $f'(x) = \frac{\mathrm{e}^{x}\arcsin\left(\sqrt{x}\right) - \frac{\mathrm{e}^{x}}{2\,\sqrt{x}\sqrt{1 - x}}}{\left(\arcsin\left(\sqrt{x}\right)\right)^{2}}$
  10. $f(x) = \frac{\ln\left(x\right)}{\tan\left(x^{3}\right)}$ Rechner
    $f'(x) = \frac{\frac{\tan\left(x^{3}\right)}{x} - \frac{3\,x^{2}\ln\left(x\right)}{\cos^{2}\left(x^{3}\right)}}{\tan^{2}\left(x^{3}\right)}$
  11. $f(x) = \frac{\mathrm{e}^{x}}{\left(x^{2} + x\right)^{3}}$ Rechner
    $f'(x) = \frac{\mathrm{e}^{x}\left(x^{2} + x\right) - 3\,\mathrm{e}^{x}\left(2\,x + 1\right)}{\left(x^{2} + x\right)^{4}}$
  12. $f(x) = \frac{\cos\left(2\,x\right)}{\sin\left(x\right)}$ Rechner
    $f'(x) = \frac{-2\,\sin\left(2\,x\right)\sin\left(x\right) - \cos\left(2\,x\right)\cos\left(x\right)}{\sin^{2}\left(x\right)}$
  13. $f(x) = \frac{\ln\left(x\right)}{x^{x}}$ Rechner
    $f'(x) = \frac{\frac{1}{x} - \ln\left(x\right)\left(\ln\left(x\right) + 1\right)}{x^{x}}$
  14. $f(x) = \frac{\mathrm{e}^{x^{2}}}{\mathrm{e}^{x}}$ Rechner
    $f'(x) = \mathrm{e}^{x^{2} - x}\left(2\,x - 1\right)$
  15. $f(x) = \frac{\ln\left(x^{2} + 1\right)}{\ln\left(x\right)}$ Rechner
    $f'(x) = \frac{\frac{2\,x}{x^{2} + 1}\,\ln\left(x\right) - \frac{\ln\left(x^{2} + 1\right)}{x}}{\left(\ln\left(x\right)\right)^{2}}$