Let's say I have a function: $$f:V\to V'$$To check if this function has derivative at $a\in V$ one can simply use the definition of derivative to see if the limit exists.
Is there a way to check if, for a given $m$, $f\in C^m(V)$ without checking if the derivative exists $m$ times?
If no for a general $m$, maybe there are special cases? Like checking if a function is smooth.
There are classes of functions which are differentiable infinitely many times.
Polynomials, such as$$ P(x)= a_0+ a_1x+a_2x^2+...a_nx^n $$
Exponentials such as $e^{kx}, e^{P(x)}$,where $P(x)$ is a polynomial.
Some trig functions such as $$\sin (kx), \cos (kx) $$
Product or composition of the above functions are also differentiable infinitely many times.
For functions which are not well known to be differentiable we have to check them out. for example $f(x) = \sqrt x$ is not differentiable at $ x=0$ so its higher derivatives do not exist at that point.