Deriving the heat equation

[JasonZ]

I just started a class called Fourier Series. The first day one of my problems is to derive the heat equation for a 1 dimensional thin rod. I can derive it no problem given the lateral surfaces are insulated. However, this problem wants me to derive it given the lateral surface transfers heat at the rate of u(x,t).

I am not exactly sure where to fit this into the equation for energy conservation. Can anyone provide some help?

Thanks,
-Jason