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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

🔥 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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