Derivative of cumulative distribution function Lets say  f is probability funktion to cdf F.
I want to compute $$lim_{ \Delta t \rightarrow 0+}\frac{ P( t<X<t +\Delta t )}{\Delta t}  $$
I should get $f(t) $. How do I get to that?

If you are assuming that $F(t)=P[X \le t]$ is differentiable, then $F$ is continuous and so $P[t < X < t+\Delta]= P[t < X \le t+\Delta]= F(t+\Delta)-F(t)$ (as long as $\Delta \ge 0$). 
(If $\Delta<0$ then $P[t < X < t+\Delta]= 0$.)
Then $\lim_{\Delta \downarrow 0} {P[t < X < t+\Delta] \over \Delta}=\lim_{\Delta \downarrow 0} {F(t+\Delta)-F(t) \over \Delta}=\lim_{\Delta \to 0} {F(t+\Delta)-F(t) \over \Delta} =F'(t)=f(t)$.
