Aus Online Mathematik Brückenkurs 2

\displaystyle f'(x)=\lim_{h \to 0}\frac{f(x+h)-f(x)}{h} wobei \displaystyle f(x)=x^2-3x+1 \displaystyle \begin{align} f'(x)&=\lim_{h \to 0}\frac{(x+h)^{2}-3(x+h)+1-(x^{2}-3x+1)}{h}\\ &=\lim_{h \to 0}\frac{x^{2}+2hx+h^{2}-3x-3h+1-x^{2}+3x+1}{h}\\ &=\lim_{h \to 0}\frac{2hx+h^{2}-3h}{h}\\ &=\lim_{h \to 0}2x+h-3\\ &=2x-3\end{align}