Ableitungsregel für Sinus

\begin{align*} {\Bigl[ \sin(x) \Bigr]}' &= \lim\limits_{h \rightarrow 0}{\left( \frac{\sin(x+h) - \sin(x)}{h} \right)} \\[0.75em] &= \lim\limits_{h \rightarrow 0}{\left( \frac{\sin(x) \cdot \cos(h) + \cos(x) \cdot \sin(h) - \sin(x)}{h} \right)} \\[0.75em] &= \lim\limits_{h \rightarrow 0}{\left( \frac{\sin(x) \cdot \left( \cos(h) - 1 \right) + \cos(x) \cdot \sin(h)}{h} \right)} \\[0.75em] &= \lim\limits_{h \rightarrow 0}{\left( \frac{\sin(x) \cdot \left( \cos(h) - 1 \right)}{h} \right)} + \lim\limits_{h \rightarrow 0}{\left( \frac{\cos(x) \cdot \sin(h)}{h} \right)} \\[0.75em] &= \sin(x) \cdot \lim\limits_{h \rightarrow 0}{\left( \frac{\cos(h) - 1}{h} \right)} + \cos(x) \cdot \lim\limits_{h \rightarrow 0}{\left( \frac{\sin(h)}{h} \right)} \\[0.75em] &= \sin(x) \cdot 0 + \cos(x) \cdot 1 \\[0.75em] &= \cos(x) \end{align*}