Suppose $\left\lbrace f_n\right\rbrace _{n=1}^{\infty}$ is a sequence of continuous functions, each of which is right differentiable on $\left[ a, b\right) $. Suppose further that for some $x_0 \in \left[ a, b\right) $ converges and that $f_i'(x+)$ converges uniformly to $g$ on $\left[ a, b\right) $. Then:
$ f_n$ converges uniformly on $\left[ a, b\right) $ to a function f ; and
$f$ is right differentiable on $\left[ a, b\right) $ and $f'(x+)=g(x)$ for all $x \in \left[ a, b\right)$.
Note: $f'(x+)$ is the right derivative of $f$
The statement is false. Consider the interval $ [a,b) = [0,2) $, and define the functions $$ f_n(x) = \begin{cases} 0 &\mbox{if } 0 \leq x < 1 \\ n & \mbox{if } 1 \leq x < 2 \end{cases} $$ These are right-differentiable with derivative $ f_n'(x+) \equiv 0 $, and converge on $ [0,1) $. The sequence is a clear counter-example to your problem as stated. Please be more careful to state all hypotheses of your problem, or indicate if you are not sure whether the statement is true or false.