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\require{AMSmath}

Bewijs van de productregel

In Almering e.a. (6de druk) lezen we:

$
\eqalign{
  & \left[ {f(a) \cdot g(a)} \right]^'  =   \cr
  & \mathop {\lim }\limits_{x \to a} \frac{{\left( {f \cdot g} \right)\left( x \right) - \left( {f \cdot g} \right)\left( a \right)}}
{{x - a}} =   \cr
  & \mathop {\lim }\limits_{x \to a} \frac{{f\left( x \right) \cdot g\left( x \right) - f\left( a \right) \cdot g\left( a \right)}}
{{x - a}} =   \cr
  & \mathop {\lim }\limits_{x \to a} \frac{{f\left( x \right) \cdot g\left( x \right) - f\left( a \right) \cdot g\left( a \right) - f\left( a \right) \cdot g\left( x \right) + f\left( a \right) \cdot g\left( x \right)}}
{{x - a}} =   \cr
  & \mathop {\lim }\limits_{x \to a} \frac{{f\left( x \right) \cdot g\left( x \right) - f\left( a \right) \cdot g\left( x \right) - f\left( a \right) \cdot g\left( a \right) + f\left( a \right) \cdot g\left( x \right)}}
{{x - a}} =   \cr
  & \mathop {\lim }\limits_{x \to a} \frac{{\left( {f\left( x \right) - f\left( a \right)} \right) \cdot g\left( x \right) + f\left( a \right) \cdot \left( {g\left( x \right) - g\left( a \right)} \right)}}
{{x - a}} =   \cr
  & \mathop {\lim }\limits_{x \to a} \frac{{f\left( x \right) - f\left( a \right)}}
{{x - a}}\mathop {\lim }\limits_{x \to a} g\left( x \right) + f\left( a \right) \cdot \mathop {\lim }\limits_{x \to a} \frac{{g\left( x \right) - g\left( a \right)}}
{{x - a}} =   \cr
  & f'\left( a \right) \cdot g\left( a \right) + f\left( a \right) \cdot g'\left( a \right) \cr}
$


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