To more understanding Under which conditions we  can  say that :
$
v'(t)\leq w'(t)
$
Implies that 
$
{v}{\mathrm{(}}{t}{\mathrm{)}}\mathrm{\leq}{w}{\mathrm{(}}{t}{\mathrm{)}}
$
Is there any explanation? .thanks very much thankful.

We have a theorem, which is a direct consequence of the Mean Value theorem:

Let $f,g$ functions defined in an interval $I$, $t_0\in I$. If
(i) $\:f(t_0)\le g(t_0)$,
(ii) $f'(t)\le g'(t)$ for all $t\in I$, $t>t_0$
then for all $t>t_0$, $\;f(t)\le g(t)$.
Furthermore, if one of the inequalities in (i) or (ii) is strict, the resulting inequality is strict.

