Kvotient regel derivasjon-bevis

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Vi har:

f(x)=u(x)v(x),f´(x)=u´(x)v(x)u(x)v´(x)(v(x))2,f´(x)=limΔx0f(x+Δx)f(x)Δx

Bevis:

$f'(x)= \lim_{\Delta x \rightarrow0} \frac{\frac{u(x+\Delta x)}{v(x+ \Delta x)} - \frac{u(x)}{v(x)}}{\Delta x} \ = \lim_{\Delta x \rightarrow0} \frac{u(x+\Delta x) \cdot v(x) - {u(x) \cdot v(x+ \Delta x)}}{\Delta x \cdot v(x+ \Delta x) \cdot v(x)} \ = \lim_{\Delta x \rightarrow0} \frac{u(x+\Delta x) \cdot v(x)- u(x) \cdot v(x) - {u(x) \cdot v(x+ \Delta x) + u(x) \cdot v(x)}}{\Delta x \cdot v(x+ \Delta x) \cdot v(x)} \ = \lim_{\Delta x \rightarrow0} ( \frac{u(x+\Delta x) - u(x){\Delta x}$