Undergrad Can the Chain Rule be Applied to Simplify Divergence in Entropy Equation?

[BeeKay]

I am looking at the derivation for the Entropy equation for a Newtonian Fluid with Fourier Conduction law. At some point in the derivation I see

\frac{1}{T} \nabla \cdot (-\kappa \nabla T) = - \nabla \cdot (\frac{\kappa \nabla T}{T}) - \frac{\kappa}{T^2}(\nabla T)^2

K is a constant and T is a scalar field. It seems obvious that there is some way to use the chain rule on the middle term to get the left and right terms, but I frankly don't exactly understand the "rules" of how to use it with the divergence. I know you can't just factor out \frac{1}{T} from the middle term, but I'm not sure how to actually simplify that middle expression. Any help is appreciated.

 

[Infrared]

I think the relevant product rule is $\nabla\cdot (fV)=(\nabla f)\cdot V+f (\nabla\cdot V)$. Try applying this with $f=1/T$ and $V=\nabla T$.

You will also want to use the chain rule for $\nabla (1/T)=-\frac{\nabla T}{T^2}$.