How do I get the speed of the movement of an analog controller?

I want to be able to get the speed (rate of movement) variable at which the player is moving the analog stick in the vertical direction (or horizontal).

For example: The faster you move the analog stick in the vertical the greater the force.

You'll have to compare the results of Input.GetAxis across two different frames.

private var lastAxis = 0.0;
function Update() {
    var axis = Input.GetAxis("Vertical");
    var velocity = (axis - lastAxis) / Time.deltaTime;
    ...
    lastAxis = axis;
}

You'll probably need some kind of smoothing or hysteresis in addition to this.