Differentiering af sammensatte funktioner

Man kan differentier en sammensat funktion på følgende måde.

LaTex: (f\ \circ\ g)(x) = f(g(x))\\ (f\ \circ\ g)'(x) = f'(g(x))\cdot g'(x)\\

Bevis

Bevis for overstående regnemetode.

LaTex: \begin{eqnarray} \frac{ \Delta y}{ \Delta x}&=&a \\ \frac{g( x+ \Delta x )-g( x )}{ \Delta x}&=&g'( x ) \\ g( x+ \Delta x )-g( x )&=& \Delta x \cdot g'( x ) \\ g( x+ \Delta x )&=& \Delta x \cdot g'( x )+g( x ) \\ \\ \frac{f( g( x )+ \Delta x )-f( g( x ) )}{ \Delta x}&=&f'( g( x ) ) \\ f( g( x )+ \Delta x )-f( g( x ) )&=& \Delta x \cdot f'( g( x ) ) \\ \\ \frac{f( g( x+ \Delta x ) )-f( g( x ) )}{ \Delta x}&=&( f \circ g )'( x ) \\ f( g( x+ \Delta x ) )-f( g( x ) )&=& \Delta x \cdot ( f \circ g )'( x ) \\ f( \Delta x \cdot g'( x )+g( x ) )-f( g( x ) )&=& \Delta x \cdot g'( x ) \cdot f'( g( x ) )= \Delta x \cdot ( f \circ g )'( x ) \\ \frac{f( g( x+ \Delta x ) )-f( g( x ) )}{ \Delta x}&=&( f \circ g )'( x ) = \frac{ \Delta x \cdot g'( x ) \cdot f'( g( x ) )}{ \Delta x} = g'( x ) \cdot f'( g( x ) )\end{eqnarray}

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