Übungen zur Produktregel mit Lösungen

✏️ Leichte Aufgaben

Fokus auf der grundlegenden Anwendung der Produktregel mit einfachen Polynomen und Funktionen.

$f(x) = x^{2}\,\sin\left(x\right)$ Rechner

$f'(x) = 2\,x\,\sin\left(x\right) + x^{2}\,\cos\left(x\right)$
$f(x) = x^{3} \cdot \mathrm{e}^{x}$ Rechner

$f'(x) = \mathrm{e}^{x}\,\left(3\,x^{2} + x^{3}\right)$
$f(x) = \left(2\,x + 1\right)\,x^{4}$ Rechner

$f'(x) = 10\,x^{4} + 4\,x^{3}$
$f(x) = \mathrm{e}^{x}\,\cos\left(x\right)$ Rechner

$f'(x) = \mathrm{e}^{x}\,\left(\cos\left(x\right) - \sin\left(x\right)\right)$
$f(x) = x\,\ln\left(x\right)$ Rechner

$f'(x) = \ln\left(x\right) + 1$
$f(x) = 5\,x^{2}\,\ln\left(x\right)$ Rechner

$f'(x) = 10\,x\,\ln\left(x\right) + 5\,x$
$f(x) = \sin\left(x\right)\,\cos\left(x\right)$ Rechner

$f'(x) = \cos\left(x\right)^{2} - \sin\left(x\right)^{2}$
$f(x) = \left(x^{2} - 1\right) \cdot \mathrm{e}^{x}$ Rechner

$f'(x) = \mathrm{e}^{x}\,\left(2\,x + x^{2} - 1\right)$
$f(x) = x^{4}\,\left(x^{2} + 3\right)$ Rechner

$f'(x) = 6\,x^{5} + 12\,x^{3}$
$f(x) = 3\,x\,\sin\left(x\right)$ Rechner

$f'(x) = 3\,\sin\left(x\right) + 3\,x\,\cos\left(x\right)$
$f(x) = \ln\left(x\right)\,x^{3}$ Rechner

$f'(x) = x^{2} + 3\,x^{2}\,\ln\left(x\right)$
$f(x) = \left(x + 1\right)\,\left(x - 1\right)$ Rechner

$f'(x) = 2\,x$
$f(x) = 2 \cdot \mathrm{e}^{x}\,x$ Rechner

$f'(x) = 2 \cdot \mathrm{e}^{x}\,\left(x + 1\right)$
$f(x) = x^{5}\,\cos\left(x\right)$ Rechner

$f'(x) = 5\,x^{4}\,\cos\left(x\right) - x^{5}\,\sin\left(x\right)$
$f(x) = 4\,x^{3} \cdot \mathrm{e}^{x}$ Rechner

$f'(x) = \mathrm{e}^{x}\,\left(12\,x^{2} + 4\,x^{3}\right)$

🔥 Mittelschwere Aufgaben

Kombination der Produktregel mit Wurzeln, negativen Exponenten und verschachtelten Funktionen.

$f(x) = \sqrt{x}\,\sin\left(x\right)$ Rechner

$f'(x) = \frac{\sin\left(x\right)}{2\,\sqrt{x}} + \sqrt{x}\,\cos\left(x\right)$
$f(x) = \frac{1}{x^{3}}\,\ln\left(x\right)$ Rechner

$f'(x) = -\frac{3\,\ln\left(x\right)}{x^{4}} + \frac{1}{x^{4}}$
$f(x) = \mathrm{e}^{-x}\,x^{2}$ Rechner

$f'(x) = \mathrm{e}^{-x}\,\left(2\,x - x^{2}\right)$
$f(x) = \left(x^{2} + 2\,x\right)\,\sqrt{x}$ Rechner

$f'(x) = \left(2\,x + 2\right)\,\sqrt{x} + \frac{x^{2} + 2\,x}{2\,\sqrt{x}}$
$f(x) = \sin\left(2\,x\right) \cdot \mathrm{e}^{x}$ Rechner

$f'(x) = \mathrm{e}^{x}\,\left(2\,\cos\left(2\,x\right) + \sin\left(2\,x\right)\right)$
$f(x) = \frac{1}{x}\,\cos\left(x\right)$ Rechner

$f'(x) = -\frac{\cos\left(x\right)}{x^{2}} - \frac{\sin\left(x\right)}{x}$
$f(x) = \ln\left(x\right)\,\tan\left(x\right)$ Rechner

$f'(x) = \frac{\tan\left(x\right)}{x} + \frac{\ln\left(x\right)}{\cos\left(x\right)^{2}}$
$f(x) = 2^{x}\,x^{3}$ Rechner

$f'(x) = 2^{x}\,x^{2}\,\left(x\,\ln\left(2\right) + 3\right)$
$f(x) = \arcsin\left(x\right)\,x$ Rechner

$f'(x) = \frac{x}{\sqrt{1 - x^{2}}} + \arcsin\left(x\right)$
$f(x) = \left(3\,x^{2} - 1\right)\,\cos\left(x\right)$ Rechner

$f'(x) = 6\,x\,\cos\left(x\right) - \left(3\,x^{2} - 1\right)\,\sin\left(x\right)$
$f(x) = \arctan\left(x\right) \cdot \mathrm{e}^{x}$ Rechner

$f'(x) = \mathrm{e}^{x}\,\left(\frac{1}{1 + x^{2}} + \arctan\left(x\right)\right)$
$f(x) = \sqrt[3]{x}\,\ln\left(x\right)$ Rechner

$f'(x) = \frac{\ln\left(x\right)}{3\,x^{\frac{2}{3}}} + \frac{1}{x^{\frac{2}{3}}}$
$f(x) = \mathrm{e}^{x}\,\left(\sin\left(x\right) + \cos\left(x\right)\right)$ Rechner

$f'(x) = 2 \cdot \mathrm{e}^{x}\,\cos\left(x\right)$
$f(x) = 2\,\ln\left(x\right)\,x$ Rechner

$f'(x) = 2 + 2\,\ln\left(x\right)$
$f(x) = x\,\log_{2}{\left(x\right)}$ Rechner

$f'(x) = \log_{2}{\left(x\right)} + \frac{1}{\ln\left(2\right)}$

🚀 Schwere Aufgaben

Hier muss die Produktregel auf Terme angewendet werden, die selbst komplexe innere Ableitungen erfordern.

$f(x) = \mathrm{e}^{2\,x}\,\sin\left(x^{2}\right)$ Rechner

$f'(x) = 2 \cdot \mathrm{e}^{2\,x}\,\sin\left(x^{2}\right) + 2\,x \cdot \mathrm{e}^{2\,x}\,\cos\left(x^{2}\right)$
$f(x) = \ln\left(x^{2} + 1\right)\,\sqrt{x}$ Rechner

$f'(x) = \frac{2\,x\,\sqrt{x}}{x^{2} + 1} + \frac{\ln\left(x^{2} + 1\right)}{2\,\sqrt{x}}$
$f(x) = \sin\left(\ln\left(x\right)\right)\,x^{3}$ Rechner

$f'(x) = x^{2}\,\cos\left(\ln\left(x\right)\right) + 3\,x^{2}\,\sin\left(\ln\left(x\right)\right)$
$f(x) = \arctan\left(2\,x\right)\,\cos\left(3\,x\right)$ Rechner

$f'(x) = \frac{2\,\cos\left(3\,x\right)}{1 + 4\,x^{2}} - 3\,\arctan\left(2\,x\right)\,\sin\left(3\,x\right)$
$f(x) = x^{2} \cdot \mathrm{e}^{x^{2}}$ Rechner

$f'(x) = 2\,x \cdot \mathrm{e}^{x^{2}}\,\left(1 + x^{2}\right)$
$f(x) = \sqrt{x^{2} + 1}\,\ln\left(x\right)$ Rechner

$f'(x) = \frac{x\,\ln\left(x\right)}{\sqrt{x^{2} + 1}} + \frac{\sqrt{x^{2} + 1}}{x}$
$f(x) = \cos\left(\mathrm{e}^{x}\right)\,x$ Rechner

$f'(x) = -x \cdot \mathrm{e}^{x}\,\sin\left(\mathrm{e}^{x}\right) + \cos\left(\mathrm{e}^{x}\right)$
$f(x) = \frac{1}{x + 1}\,\sin\left(x\right)$ Rechner

$f'(x) = -\frac{\sin\left(x\right)}{\left(x + 1\right)^{2}} + \frac{\cos\left(x\right)}{x + 1}$
$f(x) = \arcsin\left(\sqrt{x}\right) \cdot \mathrm{e}^{x}$ Rechner

$f'(x) = \frac{\mathrm{e}^{x}}{2\,\sqrt{x}\,\sqrt{1 - x}} + \mathrm{e}^{x}\,\arcsin\left(\sqrt{x}\right)$
$f(x) = \tan\left(x^{3}\right)\,\ln\left(x\right)$ Rechner

$f'(x) = \frac{3\,x^{2}\,\ln\left(x\right)}{\cos\left(x^{3}\right)^{2}} + \frac{\tan\left(x^{3}\right)}{x}$
$f(x) = \left(x^{2} + x\right)^{3} \cdot \mathrm{e}^{x}$ Rechner

$f'(x) = \mathrm{e}^{x} \cdot \left(x^{2} + x\right)^{2}\,\left(3\,\left(2\,x + 1\right) + x^{2} + x\right)$
$f(x) = \sin\left(x\right)\,\cos\left(2\,x\right)$ Rechner

$f'(x) = \cos\left(x\right)\,\cos\left(2\,x\right) - 2\,\sin\left(x\right)\,\sin\left(2\,x\right)$
$f(x) = x^{x}\,\ln\left(x\right)$ Rechner

$f'(x) = x^{x}\,\left(\ln\left(x\right)^{2} + \ln\left(x\right) + \frac{1}{x}\right)$
$f(x) = \mathrm{e}^{x + x^{2}}$ Rechner

$f'(x) = \mathrm{e}^{x + x^{2}}\,\left(1 + 2\,x\right)$
$f(x) = \ln\left(x\right)\,\ln\left(x^{2} + 1\right)$ Rechner

$f'(x) = \frac{\ln\left(x^{2} + 1\right)}{x} + \frac{2\,x\,\ln\left(x\right)}{x^{2} + 1}$