I'm working on understanding Lorentz transformations via a text by Garrity, "Electricity and Magnetism for Mathematicians: A Guided Path from Maxwell's Equations to Yang-Mills". On pages 43 and 44 he describes how to calculate the velocity of a particle in a second reference frame, with a transformation between the first and second reference frames given by a Lorentz transformation.
I've written my question below and taken a picture. My difficulty is understanding the dependencies of the different variables, and also taking the derivatives. Please help? Thanks!
$\frac{dx_1}{dt_1}$ would be the velocity as observed in the first coordinate frame. $\frac{dx_2}{dt_2}$ would be the velocity as observed in the second coordinate frame, which is what we are after.
The chain rule for your second formulation is for partial derivatives ($x_2$ and $t_2$ being functions of at least $x_1$ and $t_1$) : $$\frac {\partial x_2}{\partial t_2}=\frac {\partial x_2}{\partial t_1}\frac {\partial t_1}{\partial t_2}+\frac {\partial x_2}{\partial x_1}\frac {\partial x_1}{\partial t_2}$$
Concerning the text you could too use the linearity of Lorentz transformation (matrix $A$) to get : \begin{align} \Delta x_2&=\gamma\,\Delta x_1-\gamma\,v\,\Delta t_1\\ \Delta t_2&=-\,\gamma \frac v{c^2}\Delta x_1+\gamma\,\Delta t_1 \end{align} so that the quotient $\;\dfrac{\left(\dfrac {\Delta x_2}{\Delta t_1}\right)}{\left(\dfrac {\Delta t_2}{\Delta t_1}\right)}$ will admit the limit : $\;\dfrac {d x_2}{d t_2}=\dfrac {\gamma\, \dfrac{dx_1}{dt_1}-\gamma\,v\, }{-\,\gamma \frac v{c^2}\,\dfrac{d x_1}{d t_1}+\gamma }\;$
Hoping this clarified things.