It is corect! since lim [f(x0+h)-f(x0-h)]/h = 2f'(x0)
to prove this get the limit definition of the derivative....lim[f(x0+h)-f(x0)]/h = f'(x0) and lim[f(x0-h)-f(x0)]/h = -f'(x0)
it is easy..
If f is differentiable, then the given limit exists and is what you say it is, so you are correct. The converse is not true, though. The limit can exist for functions that are not differentiable - take and , for example.