Für $f(x)=x^2$:

\[ f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h} = \lim_{h \to 0} \frac{(x+h)^2-x^2}{h} = \\ \lim_{h \to 0} \frac{x^2+2hx+h^2 -x^2}{h} = \lim_{h \to 0} \frac{2hx+h^2}{h} = \\ \lim_{h \to 0} \frac{h(2x+h)}{h} = \lim_{h \to 0} (2x+h) = 2x \]

Analog für $f(x)=x^3$: \[ f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h} = \lim_{h \to 0} \frac{(x+h)^3-x^3}{h} = \\ \lim_{h \to 0} \frac{x^2+3hx^2+3h^2x+h^3 -x^3}{h} = \lim_{h \to 0} \frac{3hx^2+3h^2x+h^3}{h} = \\ \lim_{h \to 0} \frac{h(3x^2+3hx+h^2)}{h} = \lim_{h \to 0} (3x^2+3hx+h^2) = 3x^2 \]