Condition for derivative sequence to converge? Let $E_n(R)$ be a sequence of function that converge to $E(R)$, When we can said that $\dot{E}_n(R)$ converge to $\dot{E}(R)$. 

I can assume:


  
  uniform converge of $E_n$ to $E$.
  $E(R)$ convex.
  Continuety of $\ddot{E}$
  Monotonicity of $E_n$ os that the example of $\frac{1}{n}\sin(nx)$ does not work.
  


I need a positive result so if other condition needed for converge, its OK to point them.

Thanks !
My smart book says:


Uniform convergence of series $\dot{E_n},\,\,x\in \langle a, b\rangle$
Convergence of series $E_n$ at least one point $\in \langle a, b\rangle $(from points 1 and 2 $\Rightarrow$ series $E_n$ convergent uniformly in the interval $\langle a, b\rangle $)


No additional conditions.
My smart book says:


Uniform convergence of series $\dot{E_n},\,\,x\in \langle a, b\rangle$
Convergence of series $E_n$ at least one point $\in \langle a, b\rangle $(from points 1 and 2 $\Rightarrow$ series $E_n$ convergent uniformly in the interval $\langle a, b\rangle $)


No additional conditions.