The first variation of a functional, $J(\rho)$, is given by
$\frac{d}{d \epsilon} J(\rho + \epsilon h) |_{\epsilon = 0},$
for functions $\rho, h$ and scalar $\epsilon$ (https://en.wikipedia.org/wiki/First_variation). Is there a chain rule for the first variation? I've only been able to find a couple, unreliable references on this subject online and haven't been able to prove this myself.