Originally Posted by dhk2 .
You improve maximum speed and maximum power output with specific training. Weightlifters train with progressive resistance by adding weights and moving them slowly. But since power is a product of both force and leg speed, you'll need to train with progressive applications of force and speed (cadence). Suggest you focus on training max leg speed (cadence) rather than just trying to push harder.
You can add a set of accelerations to your normal ride, say 5-10 short intervals, with recovery spinning time in between. Pick a comfortable gear, say 39/19, and accelerate gradually and smoothly in the saddle up to a cadence of 130-140 or higher. Hold the cadence for a few seconds, then back off when you start getting winded. You'll know right away if you picked a gear that's too easy or too hard. If you can do that kind of cadence smoothly in your starting gear, go up a cog for the next interval, When you arrive at a gear that's tough, ie, you can only reach say 100 rpm rather than your full cadence, that's the one you need for your training.