I don't profess to understand the entire derivation of the equations but they are beautiful in their simplicity.
The jumble of terms after "t" is the key.
As the relative velocity "v" tends towards the speed of light "c" the whole lot tends towards 0, i.e. SQRT(c^2 - c^2) = 0
I graphed the change to get a feel for the rate of change - rather than trying to differentiate it
It starts off as a very small effect, you have to be going 30% the speed of light (90,000 km/s) before you see about a 5% change in time, length or mass (compared to the stationary reference frame). No wonder Newton did not notice it!