You would have to take this drawing
and extend the two diagrams towards the bottom (negative t). Then get the light-beams going from E0/E0' to cross the blue lines(you also extend towards the bottom).
Call the events E3/E3' and E4/E4'.
What you want to get is a formula which let's you see by which factor MOVING clocks would run slower, seen from an observer in an inertial frame of reference, compared to clocks being at rest inside the observer's frame.
All clocks following the blue line in system A, are moving at vrel = 0.5c in system A. In system B, they are at rest.
getting the Δt/Δt' between E1/E3 and E1'/E3'
Do the same for system B. This time we need a clock which moves at vrel=0,5c relative to an observer being at rest in system B. This would be the green line. After extending the diagram to the bottom along with the green line, cross some light-beams that go through E0' with the green lines.
E5/E5' and E6/E6' where the beams cross the green lines.
get the formulas for the Δt2/Δt2' and you should be able to solve similar to how i solved for the length contraction.(you only need either E3/E3'/E5/E5' OR alternatively E4/E4'/E6/E6' to arrive at the formulas)
Just remember that because of one of the two postulates of SR:
"The laws of physics are the same in all inertial frames of reference." implying that,
by whatever factor (γ) Δt is modified to arrive at Δt' using Δt * γ = Δt'
the same is true the other way around Δt2' * γ = Δt2I hope this is good enough to get you started. The description is sloppy but that is because it's harder to imagine than measuring distances with rulers. The clocks here are used just like rulers, to measure time-frames. But before i fry my brain, trying to explain this further, i am out of here.Warning: I call the two postulates, Axioms. Postulates is the better term here i believe. Also, the factor γ i use is inverse to the γ you will find on Wikipedia. This is because of the way i derived the formula, where using γ that way made perfect sense.
If you substitute γ in my formulas with sqrt(1-(v^2/x^2)), you arrive at the same formulas as Wikipedia does, so there is not really anything wrong here other than maybe not going by the standard or being a carbon copy of Wikipedia.
Also noteworthy. The numbering of the two Axioms/Postulates of SR as i used it, is flipped compared to how Wikipedia has it. You might want to change that in your derivation to be more in accord with the standard. Again, it seemed more reasonable to me to flip this, and goes along to how i derived it as i used, light always traveling at C in a vacuum absent of gravity seen from any arbitrary inertial reference frame, for the first part of my derivation.
