We all know that a primitive $F(x)$ of a function $f(x)$ is the function which derivative is $f(x)$. For example:
$$\int 2x dx = x^2 + C$$
where $x^2 + C$ is the set of all primitives of the function. However, how can I be sure that $x^2 + C$ is the only possible solution? How can I be sure that a function $f(x) \ne x^2 + C\ ,\ f'(x) = 2x$ does not exist?
How could I create a mathematical proof of this fact (in this specific case and in the general case)? Does it make sense to seek such a proof? If not, why?