Übungen zur Summenregel mit Lösungen

✏️ Leichte Aufgaben

Fokus auf der grundlegenden Anwendung der Summen- und Potenzregel bei Polynomen.

$f(x) = x^{2} + x$ Rechner

$f'(x) = 2\,x + 1$
$f(x) = x^{3} - 5\,x$ Rechner

$f'(x) = 3\,x^{2} - 5$
$f(x) = 4\,x^{2} + 2\,x + 1$ Rechner

$f'(x) = 8\,x + 2$
$f(x) = x^{5} - 3\,x^{4}$ Rechner

$f'(x) = 5\,x^{4} - 12\,x^{3}$
$f(x) = 2\,x + 7$ Rechner

$f'(x) = 2$
$f(x) = x^{3} + x^{2} - 10$ Rechner

$f'(x) = 3\,x^{2} + 2\,x$
$f(x) = -2\,x^{4} + 6\,x^{2}$ Rechner

$f'(x) = -8\,x^{3} + 12\,x$
$f(x) = x^{6} - x$ Rechner

$f'(x) = 6\,x^{5} - 1$
$f(x) = 10\,x + 3\,x^{2}$ Rechner

$f'(x) = 10 + 6\,x$
$f(x) = x^{7} - 7\,x$ Rechner

$f'(x) = 7\,x^{6} - 7$
$f(x) = 9\,x^{8} + 4\,x^{3}$ Rechner

$f'(x) = 72\,x^{7} + 12\,x^{2}$
$f(x) = 5 - x^{2}$ Rechner

$f'(x) = -2\,x$
$f(x) = x^{10} + x^{9}$ Rechner

$f'(x) = 10\,x^{9} + 9\,x^{8}$
$f(x) = 6\,x^{3} - 2\,x^{2} + 8$ Rechner

$f'(x) = 18\,x^{2} - 4\,x$
$f(x) = -x^{4} + 3\,x^{3} - x$ Rechner

$f'(x) = -4\,x^{3} + 9\,x^{2} - 1$

🔥 Mittelschwere Aufgaben

Kombination der Summenregel mit Wurzeln, Brüchen und anderen grundlegenden Funktionen.

$f(x) = \sqrt{x} + 3\,x^{2}$ Rechner

$f'(x) = \frac{1}{2\,\sqrt{x}} + 6\,x$
$f(x) = \mathrm{e}^{x} - x^{4}$ Rechner

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

$f'(x) = \cos\left(x\right) - \sin\left(x\right)$
$f(x) = 4\,\ln\left(x\right) - 2\,x$ Rechner

$f'(x) = \frac{4}{x} - 2$
$f(x) = \frac{1}{x^{2}} + 5\,x^{3}$ Rechner

$f'(x) = -\frac{2}{x^{3}} + 15\,x^{2}$
$f(x) = \frac{1}{x} + \sqrt[3]{x}$ Rechner

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

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

$f'(x) = -\frac{3}{x^{2}} + \frac{2}{\sqrt{x}}$
$f(x) = \tan\left(x\right) - 5 \cdot \mathrm{e}^{x}$ Rechner

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

$f'(x) = 1 + \cos\left(x\right)$
$f(x) = \arcsin\left(x\right) + \arccos\left(x\right)$ Rechner

$f'(x) = 0$
$f(x) = 10\,x^{5} + 2^{x}$ Rechner

$f'(x) = 50\,x^{4} + 2^{x}\,\ln\left(2\right)$
$f(x) = 3\,\cos\left(x\right) - 2\,\sin\left(x\right)$ Rechner

$f'(x) = -3\,\sin\left(x\right) - 2\,\cos\left(x\right)$
$f(x) = \arctan\left(x\right) - \frac{1}{x}$ Rechner

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

$f'(x) = \frac{1}{x} + \mathrm{e}^{x} + 1$

🚀 Schwere Aufgaben

Hier muss die Summenregel auf Terme angewendet werden, die selbst die Produkt-, Quotienten- oder Kettenregel erfordern.

$f(x) = x^{2}\,\sin\left(x\right) + 5\,x^{3}$ Rechner

$f'(x) = 2\,x\,\sin\left(x\right) + x^{2}\,\cos\left(x\right) + 15\,x^{2}$
$f(x) = \frac{x}{x + 1} - \mathrm{e}^{x}$ Rechner

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

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

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

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

$f'(x) = \mathrm{e}^{x} + x \cdot \mathrm{e}^{x} + \frac{1}{x^{2}}$
$f(x) = \sqrt{x^{2} + 4} + \sin\left(x\right)$ Rechner

$f'(x) = \frac{x}{\sqrt{x^{2} + 4}} + \cos\left(x\right)$
$f(x) = \arctan\left(x\right) - x\,\ln\left(x\right)$ Rechner

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

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

$f'(x) = \frac{x\,\cos\left(x\right) - \sin\left(x\right)}{x^{2}} + 2\,x$
$f(x) = 5 \cdot \left(x^{2} + 3\right)^{4} - 2\,x$ Rechner

$f'(x) = 40\,x \cdot \left(x^{2} + 3\right)^{3} - 2$
$f(x) = \ln\left(\sin\left(x\right)\right) + \mathrm{e}^{x}$ Rechner

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

$f'(x) = \frac{2}{\sqrt{1 - 4\,x^{2}}} + \frac{1}{\sqrt{1 - x^{2}}}$
$f(x) = x^{x} + 5$ Rechner

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

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