It's not a good idea to try to make up time on the road when you are late. We once called this behavior "pressure driving." Pressure driving is characterized by speeding, sudden and multiple lane changes, cutting other drivers off, tailgating, or going on private property or off the road to get around obstacles or slower drivers. In short, it's acting in an intimidating manner toward other drivers. It's an example of human beings at their worst (my opinion).
In some places, it can also be a criminal act. In my state, most traffic offenses are civil, meaning they are handled in court more informally, and you cannot be arrested for them. Criminal acts can result in arrest and detention, and they are handled by attorneys in more formal court proceedings. What driving instructors used to call pressure driving is now called "aggressive driving" under Arizona law, and it is a serious criminal offense.
What's the point, anyway? In most cities, on city streets or boulevards, the traffic lights are "timed." For example, in theory, if a street is timed at 45 mph, and you start from a new green light, accelerate promptly to 45 mph and hold that speed, provided you have no traffic in front of you, you would hit every succeeding light green for as long as the street remains "set" at 45 mph. This also means that if you drive faster than 45 mph, you will encounter almost every red light that is possible on that street! It is therefore almost impossible to make up time on city streets -- you might make it through one light, but after that you're going to hit a lot of red ones. Traffic considerations on city expressways also limit how much time you can make up when you are in a hurry.
It is far better to allow yourself an early start so you can get to your destination without having to rush. When that's not possible, and you are unavoidably late, or traffic is extraordinarily snarled, then relax and recognize that getting impatient and making driving errors that may cause you to have a collision won't really save you any time. If you're going to be late anyway, does your boss really care if it is seven minutes or nine minutes? Late is late, so there's no point in getting more stressed about it.
I don't know about you, but I've done some really stupid things behind the wheel when I've been in a hurry. If I'm ever rude to someone, it's going to be in those circumstances where I'm rushed or impatient and someone does something that causes me to lose a few seconds. Over many years of driving, I have regularly seen that even though someone passed me in a huge hurry, they don't often get anywhere. I'm usually still right with them when they take that off-ramp or make a turn, miles down the road -- or when they reach that next red light. So for me, it makes sense to practice being patient. Since I have to share the road with YOU, I obviously hope you feel the same way.
Keep it between the fence posts!