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### 13 Differentialrechnung

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#### 13.2 Ableitungsregeln

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##### 13.2.6 Logarithmische Ableitung

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Beispiel 13 - 79
$y=\frac{u}{v}$ .

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$y=\frac{u}{v}$ (Produktregel)

 $f\left(x\right)$ $=$ $\frac{u\left(x\right)}{v\left(x\right)}$ $lnf\left(x\right)$ $=$ $ln\left(\frac{u\left(x\right)}{v\left(x\right)}\right)$ $=$ $lnu\left(x\right)-lnv\left(x\right)$ ${f}^{\prime }\left(x\right)\cdot \frac{1}{f\left(x\right)}$ $=$ $\frac{1}{u\left(x\right)}\cdot {u}^{\prime }\left(x\right)-\frac{1}{v\left(x\right)}\cdot {v}^{\prime }\left(x\right)$ $=$ $\frac{{u}^{\prime }\left(x\right)\cdot v\left(x\right)-u\left(x\right)\cdot {v}^{\prime }\left(x\right)}{u\left(x\right)\cdot v\left(x\right)}$ ${f}^{\prime }\left(x\right)$ $=$ $\frac{1}{u\left(x\right)}\cdot {u}^{\prime }\left(x\right)-\frac{1}{v\left(x\right)}\cdot {v}^{\prime }\left(x\right)$ $=$ $\frac{{u}^{\prime }\left(x\right)\cdot v\left(x\right)-u\left(x\right)\cdot {v}^{\prime }\left(x\right)}{u\left(x\right)\cdot v\left(x\right)}\cdot \frac{u\left(x\right)}{v\left(x\right)}$ $=$ $\frac{1}{u\left(x\right)}\cdot {u}^{\prime }\left(x\right)-\frac{1}{v\left(x\right)}\cdot {v}^{\prime }\left(x\right)$ $=$ $\frac{{u}^{\prime }\left(x\right)\cdot v\left(x\right)-u\left(x\right)\cdot {v}^{\prime }\left(x\right)}{u\left(x\right)\cdot v\left(x\right)}\cdot \frac{u\left(x\right)}{v\left(x\right)}$

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