Constantly accelerating rocket algebra problem

[Fek]

Homework Statement

• Rocket is accelerating constantly. Let S' be instantaneous rest frame of rocket and S be frame in which rocket is observed moving at velocity v.

Homework Equations

Given: $$ dv = dv' (1 - v^2) $$

Must prove:
$$ \frac{dv}{dt} = \frac{dv'}{dt'} (1 - v^2)^{\frac{3}{2}} $$

The Attempt at a Solution

So differentiate given equation with respect to t and use chain rule to get in terms of t'

$$ \frac{dv}{dt} = \frac{dt'}{dt} * \frac{d}{dt'}[dv'(1 - v^2)] $$
We also know
$$ \frac{dt'}{dt} = (1 - v^2)^{\frac{1}{2}} $$
as t' is proper time.

We also have:
$$ \frac{d}{dt'} (dv' (1 - v^2) = \frac{dv'}{dt'} (1 - v^2) $$

We have the answer! Except this final step isn't right because v is a function of t' as well and chain rule must be used?

 

[PeroK]

If you have a functional equation, you can differentiate it. If you have an equation involving infinitesimal differentials, you can't differentiate it. Instead, you can divide by another infinitesimal differential.