The question comes from S. Farlow's book on PDEs and says: Derive the heat equation $$ u_t(x,t) = \frac{1}{cp}\frac{\partial}{\partial x}[k(x)u_{x}(x,t)] + f(x,t) $$ , for the situation where the thermal conductivity $k(x)$ depends on $x$.
As in the text, I've done the following:
Net change of heat inside $[x,x+\Delta x]$ = Net flux of heat across boundaries + Total heat generated inside $[x,x+\Delta x]$
$$\textit{Total Heat Inside} [x,x+\Delta x]= cpA \int _{ x}^{x+\Delta x}u(s,t) ds $$
$$ \frac{d}{dt} \int _{ x}^{x+\Delta x} c\rho A u(s,t) ds = c\rho A \int _{ x}^{x+\Delta x} u_t(s,t) ds $$ Here is where I made the change $$ = A [ k(x+\Delta x)u_x(x+\Delta x,t) - k(x)u_x(x,t)] +A \int _{x}^{x+\Delta x} f(s,t) ds $$
After applying the Mean Value Theorem as in the lecture, I get:
$$ c\rho A u_t(\xi_1,t)\Delta x = A[k(x+\Delta x)u_x(x+\Delta x, t) - k(x)u_x(x,t)] + Af(\xi_2,t)\Delta x $$
$$ x < \xi < x+\Delta x $$
Thus, I end up with:
$$ u_t(\xi,t) = \frac{1}{c\rho} \bigg[ \frac{k(x+\Delta x)u_x(x+\Delta x,t) - k(x)u_x(x,t)} {\Delta x} \bigg] + \frac{ 1}{c\rho}f(\xi,t) $$
Thus, to get to the answer, I am supposed to take $$ \Delta x \to 0 $$, but I don't see how I can proceed further using what I have above, since I am supposed to have a "$k'(x)u_x(x,t)$" term and a $k(x)u_xx(x,t)$ term from the product rule. Is there anything I am missing? Also, I am noticing the units for the source function are not quite tying out, as I get $\frac{ 1}{c\rho}f(x,t)$, not sure why.
Now just put $$\frac{k(x+\Delta x)u_x(x+\Delta x,t) - k(x)u_x(x,t)} {\Delta x} = \frac{[k(x+\Delta x)-k(x)]u_x(x+\Delta x,t)+k(x)[u_x(x+\Delta x,t)-u_x(x,t)]}{\Delta x},$$ and take the limit; this recovers the product rule you are looking for.