I'm trying to derive the relativistic velocity addition equation using the time dilation equations but I get a wrong result.
Assume the widely used scenario in which there are 3 clocks: O, A and B.
A is moving relative to O at velocity $v$.
B is moving relative to A at velocity $w$.
Find the velocity of B relative to O (in O's frame of reference).
The time dilation equations says:

$$t\_A = t\_O \sqrt{1-v^2/c^2}$$

$$t\_B = t\_A \sqrt{1-w^2/c^2}$$

$$t\_B = t\_O \sqrt{1-u^2/c^2}$$

From these 3 equations we want to derive $u$. Then we get:

$$t\_O \sqrt{1-u^2/c^2} = t\_O \sqrt{1-v^2/c^2} \sqrt{1-w^2/c^2}$$

$$1-u^2/c^2 = (1-v^2/c^2) (1-w^2/c^2)$$

$$1-u^2/c^2 = 1-v^2/c^2-w^2/c^2+v^2 w^2/c^4$$

$$u^2 = v^2 + w^2 - v^2 w^2/c^2$$

This is different from the equation

$$u = \frac{v + w}{1+v w/c^2}$$

Why doesn't the derivation of velocity using the time dilation equations alone work?