Consider a sequence of functions $f_n$ where $f_n : \mathbb{R} \to \mathbb{R}$ and $f_n$ are all differentiable with derivatives $f^\prime_n$. The sequence $f_n$ and the sequence $f^\prime_n$ both converge uniformly to functions $f$ and $g$ respectively. According to the definition given in this wiki page on uniform convergence in section 'To Differentiability"
If $ f_n $ converges uniformly to $ f $, and if all the $ f_n $ are differentiable, and if the derivatives $f^\prime_n$ converge uniformly to $g$, then $ f $ is differentiable and its derivative is $g$.
Q.1) Can a similar definition be used for higher order derivatives ?
This is a different definition of derivative which is not same as the usual definition f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}.
Q.2) In case of both not being in agreement with each other which one should I use for further analysis on $f$ for example in theorems involving the derivative of $f$ ?