Suppose instead of defining the derivative as $$f'(x) = \lim_{h\to 0}\frac{f(x + h) - f(x)}{h}$$ we define it as $$f_q'(x) = \lim_{q\to 1}\frac{f(qx) - f(x)}{qx - x}.$$
Does this alternative definition still satisfy the linear approximation characterization of the derivative, i.e. do we have $$f(x + h) = f(x) + f_q'(x)h + o(h)$$ as we do with $f'(x)$?
I understand some care must be taken at $x = 0$.