I am trying to emulate a change of variables performed in A model of physiologically structured population dynamics with nonlinear individual growth rate (Calsina, Saldaña) in Equation (8). The idea is they model a size-structured population with a PDE. They find a solution to a PDE using the Method of Characteristics, then integrate the solution with respect to the size-variable $x$ to compute the total population.
They define $x(t)=\varphi(t;x_0,t_0)$ to be the characteristic solution to the ODE: $$x'(t)=\gamma(x(t),t)$$ $$x(t_0)=x_0$$
then make a change of variables in Equation (8) from $x \rightarrow \xi :=\varphi(0;x,t)$. They track the characteristic solution $\varphi$ that goes through the point $(x,t)$ back to the initial point $(\xi,0)$, but I am unsure how to find the partial derivative of $x$ with respect to $\xi$. My first attempt was to consider (where $x(t):=X(t,\xi)$):
\begin{align} \frac{\partial}{\partial \xi} \left(\frac{\partial X(t,\xi)}{\partial t} \right) & = \frac{\partial}{\partial \xi} \left( \gamma(X(t,\xi),t) \right) \\ & = \frac{\partial \, \gamma(X(t,\xi),t)}{\partial X} \, \frac{\partial X(t,\xi)}{\partial \xi} \end{align}
but from here I am stuck. Please help with any insight you may have.