The Conventional (Fake) Derivation of Einstein's
Mass-Velocity Equation.

General.
The "Conventional" method (below, source unknown) of deriving the relation between the original mass and the increased mass in terms of speed does not assume a constant speed of light in its derivation, yet it produces the same equation as Einstein's invalid method :-
M/Mo = (1 - V² / c²)
and so must also be faulty.

The Derivation.
With the usual notations, using the subscript zero to designate "at rest" values, the conventional method is as follows:-
energy                                             E = Mc²                                                   (1)
momentum                                       p = MV                                                   (2)
from (1) and (2),                             E = pc² / V                                               (3)
For an energy change, (where dE, dx, dp, dt are partial differentials, F is a Force, x and t are distance and time),
dE = Fdx = dp.dx/dt = V.dp                                                    (4)
Multiply (3) by (4),                    EdE = c²p.dp                                                 (5)
Integrate (5),                                  E² = c²p² + Eo²                                          (6)
where Eo is the integrating constant for initial conditions (V = 0)
From (3),                                         cp = EV / c                                                (7)
from (7) and (6),                            E² = E²V² / c² + Eo²                                   (8)
giving,                           (1 - V² / c²) = Eo²                                                     (9)
whence                     E / Eo = M / Mo = (1 - V²/c²)                                     (10)

The Flaw
When the method was examined the flaw was found to be that the equation for momentum was incorrect. It must be obvious that when energy is applied to cause unhindered motion the speed is a maximum as all the added energy appears as KE and none can be adsorbed by the moving object, so the mass stays constant at Mo. Therefore the equation for momentum should be p = MoV, not MV . To eliminate Mo requires the use of equation Eo = Mo, instead of eq.(1), and these are the first two equations of this method. Following the same method used, with the corrected equations, leads to an equation identical to NR eq.(13) - the first order approximation of eq.10 above. If mass M is used (as it was), one of the partial differentials is not valid as M is assumed to be a constant there. It becomes valid when Mo is used.