Why $(f(-x)) ^\prime=-f ^ \prime(-x)$? Why this equaility holds $(f(-x)) ^\prime=-f ^ \prime(-x)$? I do not get it utterly.

By the chain rule we have: $$f(g(x))'=f'(g(x))\cdot g'(x)$$ Set $g(x)=-x$. Then $g'(x)=-1$...