$\displaystyle \int_{x}^{x+\Delta x} \frac{\partial e}{\partial t}(\xi,t)d\xi $ | $\displaystyle = \frac{d}{dt} \int_{x}^{x+\Delta x} e(\xi,t)d\xi $ | by Leibniz's rule |
$\displaystyle = -\bigl(\phi\bigl(x+\Delta x,t\bigr) - \phi(x,t)\bigr) \quad $ | by the conservation of heat energy |