Die lokale Änderungsrate ist
\(\begin{aligned}l'(t) & =\lim_{h\to0}\frac{l(t+h)-l(t)}{h}\\ & =\lim_{h\to0}\frac{\frac{9,81}{2}(t+h)^{2}-\frac{9,81}{2}t^{2}}{h}\\ & =\lim_{h\to0}\frac{\frac{9,81}{2}t^{2}+\frac{9,81}{2}\cdot2th+\frac{9,81}{2}h^{2}-\frac{9,81}{2}t^{2}}{h}\\ & =\lim_{h\to0}\frac{\frac{9,81}{2}\cdot2th+\frac{9,81}{2}h^{2}}{h}\\ & =\lim_{h\to0}\frac{\left(\frac{9,81}{2}\cdot2t+\frac{9,81}{2}h\right)h}{h}\\ & =\lim_{h\to0}\left(\frac{9,81}{2}\cdot2t+\frac{9,81}{2}h\right)\\ & =\frac{9,81}{2}\cdot2t\\ & =9,81t\end{aligned}\)
