Trying to derive the infinitesimal time dilation relation $dt=\gamma d\tau $, where $\tau $ is the proper time, 𝑡 the coordinate time, and $\gamma =(1-v(t{)}^{2}/{c}^{2}{)}^{-1/2}$ the time dependent Lorentz factor. The derivation is trivial if one starts by considering the invariant interval $d{s}^{2}$, but it should be possible to obtain the result considering only Lorentz transformations. So, in my approach I am using two different reference frames $(t,x)$ will denote an intertial laboratory frame while $({t}^{\prime},{x}^{\prime})$ will be the set of all inertial frames momentarily coinciding with the observed particle, i.e. the rest frame of the particle. These frames are related by

${t}^{\prime}=\gamma (t-\frac{Vx}{c}),\phantom{\rule{1em}{0ex}}{x}^{\prime}=\gamma (x-Vt),$

where $V$ is some nonconstant (i.e. time dependent) parameter which is, hopefully, the velocity of the particle in the laboratory frame. Treating $x$, $t$ and $V$ as independent variables (for now) and taking the differential of the above relations,

$d{t}^{\prime}=\gamma (dt-\frac{Vdx}{c})-\frac{{\gamma}^{3}}{{c}^{2}}(x-Vt)dV,$ and

$d{x}^{\prime}=\gamma (dx-Vdt)-{\gamma}^{3}(t-\frac{Vx}{c})dV.$

Imposing either the definition of the rest frame $d{x}^{\prime}=0$ or (what should be equivalent) $dx=Vdt$, the only way in which i obtain $dt=\gamma d{t}^{\prime}$ is if $dV=0$. So, the derivation breaks badly at some point or I must be wrong in using some of the above equations. Which one is it?