Please provide me with a proof for the formula : $R(T) = \rho (1 + αT)$ which relates the change of resistivity with temperature Here, $\rho$ is the resistance at temperature ($T$) and ($R$) is the resistance at a given temperature. Also, will the value of $α$ become negative if the resistor is cooled?
 A: The equation is just a generalization of an experimental fact that resistance is weakly dependent on temperature. Assuming that $R(T)$ is an unknown function,
we can expand
$$R(T)\approx R(T_0) + R'(T_0)(T-T_0) + \mathcal{O}\left(\frac{1}{2}R''(T_0)(T-T_0)^2\right).$$
The last term is known experimentally to be small for most materials, which is why a good rule to use in not to extreme conditions is
$$ R(T)\approx R(T_0) + R'(T_0)(T-T_0) = R(T_0)\left[1 + \frac{R'(T_0)}{R(T_0)}(T-T_0)\right],$$
we can now define $\rho = R(T_0)$ and $\alpha =\frac{R'(T_0)}{R(T_0)}$ recovering the formula
$$R(T) = \rho\left[1+\alpha(T-T_0)\right].$$
Update
This is just one example of many empirical laws which are grounded in observations, but do not really have rigid status of the laws if physics. Most of them are justifiable to a good accuracy in linear regime.
To add more such examples:

*

*Ohm's law

*Relation $\mathbf{D}=\epsilon\mathbf{E}$

*Relation between the resistance and the cross-section and length of the wire: $R=\rho l/S$
The first two break already for non-linear macroscopic materials, and all three fail at nanoscale and/or in quantum regime.
An interesting counter example is the *laws of thermodynamics, which, although approximate, hold to a very high accuracy, determined by the number of molecules.
