Question: Why I can write formula derivative $$ f'(x) = \lim\_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}$$ in this form: $$ f'(x)=\frac{f(x+h)-f(h)}{h}+O(h)$$
I know, that it's easy but unfortunately I forgot.

Answer: 
Let $f(x)=|x|^{3/2}$. Then $f'(0)=0$. But $\dfrac{|h|^{3/2}-0}{h}$ is not $O(h)$.
By the definition of differentiability 
$$f(x+h)-f(x)=f'(x)h+o(h),\,\,h\to{0}.$$
In your's second formula must be $o(1),$ not $O(h).$