Calculus and Indifference Curves in an Urban Economics Example

I am reading the paper ‘The Structure of Urban Equilibria‘ by Jan Brueckner.

It uses a monocentric city model, where all consumers earn income $y$ at the centre of the city. They buy $q$ housing for a price $p$ at distance $x$ from the centre, incurring transport costs $tx$.

Consumers have a utility function:

$v(c,q)=v(y – tx – p(phi)q(phi),q(phi))=u$

where $phi=x,y,t,u$

The budget constraint is:

$c = y – tx – pq$

The tangency condition implies:

$frac{v_1(y – tx – pq, q)}{v_2(y – tx – pq, q)} = p$

where the subscript 1 denotes partial differentiation w.r.t. the first argument etc.

The paper then discusses how $p$ and $q$ vary with $x, y, t$ and $u$.

If $phi=x,y,t$, we stay on the same indifference curve. I find it relatively straightforward to find $frac{partial{p}}{partial{x}},frac{partial{p}}{partial{y}}$ and $frac{partial{p}}{partial{t}}$.

If $eta$ is the slope of the income-compensated demand curve, then $frac{partial{q}}{partial{phi}} = etafrac{partial{p}}{partial{phi}}$.

Now to allow $u$ to vary. The budget constraint swings out to meet a new indifference curve, determining the new $p$ and $q$.

I can find $frac{partial{p}}{partial{u}}$. Totally differentiate the utility function w.r.t u:

$frac{d}{du}[v(y – tx – p(phi)q(phi),q(phi))= u] = v_1(-frac{partial{p}}{partial{u}}q-pfrac{partial{q}}{partial{u}})+v_2(frac{partial{q}}{partial{u}})=1$

Since, by the tangency condition $v_2=pv_1$:

$v_1(-frac{partial{p}}{partial{u}}q-pfrac{partial{q}}{partial{u}}+pfrac{partial{q}}{partial{u}})=v_1(-frac{partial{p}}{partial{u}}q)=1$

So $frac{partial{p}}{partial{u}} = frac{-1}{qv_1}$.

The paper then quotes:

$frac{partial{q}}{partial{u}} = [frac{partial{p}}{partial{u}}-frac{partial{MRS}}{partial{c}}frac{1}{v_1}]eta$

I don’t know how to derive this. I’m guessing the first term in the square brackets is a substitution effect and the second term is an income effect.

Please help me understand this last expression $frac{partial{q}}{partial{u}} = [frac{partial{p}}{partial{u}}-frac{partial{MRS}}{partial{c}}frac{1}{v_1}]eta$ and how to derive it.