$\begin{array}{l}\frac{d}{dx}\left(\cos(kx)\right)&=&\lim_{h\rightarrow 0}\frac{\cos(k(x+h))-\cos(kx)}{h}\\&=&\lim_{h\rightarrow 0}\frac{\cos(kx)\cos(kh)-\sin(kx)\sin(kh)-\cos(kx)}{h}\\&\approx&\lim_{h\rightarrow 0}\frac{\cos(kx)\times 1-\sin(kx)\times kh-\cos(kx)}{h}\\&=&\lim_{h\rightarrow 0}-k\sin(kx)\\&=&-k\sin(kx)\end{array}$

Note that we have used $\cos(kh)\rightarrow 1$ and $\sin(kh)\rightarrow kh$ as $h\rightarrow 0$.