Es sei f definiert durch
f(x)=Ax^2+Bx +C.
Bestimmen Sie die Funktionsgleichung von f'.
\color{red}f'(x)
=
2*Ax+B
Es gilt nach Definition an einer Stelle
{\color{blue}x_0}:
\displaystyle
f'({\color{blue}x_0}) =
\lim_{x \to {\color{blue}x_0}}\,
\frac{f(x)-f( {\color{blue}x_0})}{x- {\color{blue}x_0}}.
Einsetzen des Funktionsterms von f ergibt:
\displaystyle
f'({\color{blue}x_0}) = \lim_{x \to {\color{blue}x_0}}\,
\frac{(Ax^2+Bx
+C)-(A\cdot
{\color{blue}x_0}^2+B\cdot
{\color{blue}x_0}+C)}{x- {\color{blue}x_0}}
\displaystyle
= \lim_{x \to {\color{blue}x_0}}\,
\frac{A
(x^2- {\color{blue}x_0}^2) +
B(x- {\color{blue}x_0})}{x- {\color{blue}x_0}}
= \lim_{x \to {\color{blue}x_0}}\,
\frac{A(x+ {\color{blue}x_0})(x- {\color{blue}x_0}) +
B(x- {\color{blue}x_0})}{x- {\color{blue}x_0}}
\displaystyle
= \lim_{x \to {\color{blue}x_0}}\,
[A(x+ {\color{blue}x_0}) + B]
= A({\color{blue}x_0}+{\color{blue}x_0}) + B
= 2*A{\color{blue}x_0}+B.
Das gilt für alle {\color{blue}x_0} in der
Definitionsmenge von f,
d.h. f' ist definiert durch f'(x) =
2*Ax+B.