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Bevis for f’ for ln(x) og vha omvendte funktioner

Bevis for f’ for ln(x) og vha omvendte funktioner. f(x) = ln(x) f’(x) =. Følgende forudsættes kendt: Dvs. en funktion sammensat med sin omvendte funktion giver x (Altså fx 1. x+2-2=x 2. = x) Formlen for diff af sammensat funktion: (f(g(x))’ = f’(g(x))*g’(x)

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Bevis for f’ for ln(x) og vha omvendte funktioner

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  1. Bevis for f’ for ln(x) og vha omvendte funktioner

  2. f(x) = ln(x) f’(x) = Følgende forudsættes kendt: Dvs. en funktion sammensat med sin omvendte funktion giver x (Altså fx 1. x+2-2=x 2. = x) Formlen for diff af sammensat funktion: (f(g(x))’ = f’(g(x))*g’(x) Omvendte funktion til ln(x) =

  3. Bevis for diff af ln(x) • Vi ved: , så: =x Begge sider differentieres *(ln(x)’ = 1 På venstre side benyttes formel for diff af sammensat funktion x*(ln(x)’ = 1 Første del giver stadig x, så (ln(x))’ = Der divideres med x på begge sider Hermed bevist, at f(x) = ln(x) f’(x) =

  4. f(x) = f’(x) = Følgende forudsættes kendt: Dvs. en funktion sammensat med sin omvendte funktion giver x (Altså fx 1. x+2-2=x 2. = x) Formlen for diff af sammensat funktion: (f(g(x))’ = f’(g(x))*g’(x) Omvendte funktion til =

  5. Bevis for diff af • Vi ved: , så: = x Begge sider differentieres 2= 1 På venstre side benyttes formel for diff af sammensat funktion = Der divideres med 2 på begge sider Hermed bevist, at f(x) = f’(x) =

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