TeX source:

\begin{aligned}&D(\cos (x))=\lim\limits_{h \rightarrow 0} \frac{\cos (x+h)-\cos (x)}{h}=\lim\limits_{h \rightarrow 0} \frac{\cos (x) \cos (h)-\sin (x) \sin (h)-\cos (x)}{h} \\&=\lim\limits_{h \rightarrow 0} \cos (x) \frac{\cos (h)-1}{h}-\sin (x) \frac{\sin (h)}{h} \longrightarrow \cos (x) \cdot(0)-\sin (x) \cdot(1)=-\sin (x)\end{aligned}