 Solution

Take a bar of constant cross-sectional area $A$, thermally insulated along its sides, with no internal heat production. Fourier's law says that the heat flux in the positive $x$ direction isThe thermal conductivity $k(x)$ is positive. On a small slice $[x,x+\Delta x]$, conservation of energy givesHere $\rho c_p$ is the volumetric heat capacity, the product of mass density and specific heat capacity. Divide by $A\Delta x$, take $\Delta x\to 0$, and use $\rho c_p=1$. The resulting heat equation isFor differentiable $k$, this is $T_t=k{T}_{xx}+k^{\prime }{T}_x$; retaining the divergence form also expresses conservation of energy correctly across an interface where $k$ jumps.