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13 Differentialrechnung

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13.1 Differenzierbarkeit einer Funktion

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13.1.2 Potenzregel

für $n \in ℕ$ gilt: .
$\frac{d}{d x} (x^{n}) = lim_{x \to 0} \frac{Δ y}{Δ x} = lim_{x \to 0} \frac{(x + Δ x^{)} n - x^{n}}{Δ x}$ .
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${(a + b)}^{n} = a^{n} + (\binom{n}{1}) a^{n - 1} b + (\binom{n}{2}) a^{n - 2} b^{2} + \dots + b^{n}$ .
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$\frac{Δ y}{Δ x} = \frac{f (x_{0} + Δ x) - f (x)}{Δ x} = \frac{(x + Δ x^{)} n - x^{n}}{Δ x}$ .
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$\frac{Δ y}{Δ x} =$ /<msup><mrow>x/</mrow><mrow >n</mrow></msup > + n 1 xn-1(Δx) + n 2 xn-2(Δx)2 + … + (Δx)n - Δx .
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$= (\binom{n}{1}) x^{n - 1} + (\binom{n}{2}) x^{n - 2} (Δ x) + \dots + {(Δ x)}^{n - 1}$ .
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$\frac{d}{d x} (x^{n}) = lim_{x \to 0} ((\binom{n}{1}) x^{n - 1} + (\binom{n}{2}) x^{n - 2} (Δ x) + \dots + {(Δ x)}^{n - 1})$ .
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$\frac{d}{d x} (x^{n}) = (\binom{n}{1}) \cdot x^{n - 1} = n \cdot x^{n - 1}$