Kvotient regel derivasjon-bevis

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$$\begin{gathered} f^{\prime }(x)=\underset{\mathrm{\Delta }x\to 0}{lim}\frac{\frac{u(x+\mathrm{\Delta }x)}{v(x+\mathrm{\Delta }x)}-\frac{u(x)}{v(x)}}{\mathrm{\Delta }x} \\ \underset{\mathrm{\Delta }x\to 0}{lim}\frac{u(x+\mathrm{\Delta }x)\cdot v(x)-u(x)}{\mathrm{\Delta }x\cdot v(x+\mathrm{\Delta }x)\cdot v(x)} \end{gathered}$$