שינויים

$$f(x)=\sin x \Rightarrow f'(x_0)=\lim_{\Delta x \to 0} \frac{\sin(x+\Delta x)-\sin x}{\Delta x} = \lim_{\Delta x \to 0} \frac{2\sin\frac{\Delta x}{2} \cos(x+\frac{\Delta x}{2}) }{\Delta x}=$$ $$\{\Delta x'=\frac{\Delta x}{2}\} = \lim_{\Delta x'\to 0} \frac{2\sin \Delta x' \cos (x+\Delta x')}{2\Delta x'} =\lim_{\Delta x'\to 0} 1\cdot \cos (x+\Delta x') = \cos (x)$$