According to the textbook I am reading (Fundamentals of Heat and Mass Transfer by Incropera), Fourier's law of thermal conduction, $q_x=-kA\frac{dT}{dx}$, is a law based on experimental evidence, not one that's derived from first principles. The book goes on to describe the experiment that the law is based on. A rod has length $\Delta x$, cross-sectional area $A$, and its end faces have a temperature difference of $\Delta T$. Holding $\Delta T$ and $\Delta x$ constant, we find that the heat conduction rate $q_x$ is directly proportional to the cross-sectional area $A$. Holding $\Delta T$ and $A$ constant, we find that $q_x$ is inversely proportional to the rod length $\Delta x$. Finally, holding $\Delta x$ and $A$ constant, we find that $q_x$ is directly proportional to the temperature difference between the rod's end faces, $\Delta T$. All of these results can be packed into the proportionality $q_x\propto A\frac{\Delta T}{\Delta x}$.
The problem I have with this is that the textbook only specifies that there is a temperature difference $\Delta T$ between the ends of the rod. It doesn't specify how the temperature varies between the ends of the rod is. Is it linear? If so, does the relation $q_x\propto A\frac{\Delta T}{\Delta x}$ only hold for a rod that has a linear temperature variation along its length?
This is just a guess but, maybe the text is implying that, since the temperature will vary linearly over infinitesimal lengths as you take the limit as $\Delta x$ goes to zero, we can just apply Fourier's law for every infinitesimal length along the rod and integrate?
