- #1

Adel Makram

- 635

- 15

## Homework Statement

How to obtain the famous formula of velocity transformation using a chain rule.

I know that there is a straightforward way by dividing ##dx## as a function of ##dx`## and ##dt`## on ##dt## which is also a function of them. But I would rather try using the chain rule.

## Homework Equations

##x=\gamma(x`+vt`)##

##t=\gamma(t`+\frac{v}{c^2}x`)##

## The Attempt at a Solution

I tried the following chain rule ##\frac{dx}{dt}=\frac{dx}{dx`}\frac{dx`}{dt`}\frac{dt`}{dt}## so ##u=\frac{dx}{dx`}\frac{dt`}{dt}u`##

The first term ##\frac{dx}{dx`}=\gamma(1+\frac{v}{u`})##

The second term ##\frac{dt`}{dt}## requires me to take a derivative of ##t`## with respect to ##t##. Here the equation, ##t=\gamma(t`+\frac{v}{c^2}x`)## will not help because I have to differentiate ##x`## with respect to ##t##.