Learning Calculus from First Principles

I never took calculus and I'm intimidated by it. Can you explain derivatives like I'm 5? And I mean literally 5, not pretend 5.

Okay! Imagine you're on a swing: At the bottom: You're going FAST At the top: You stop for a moment In between: You're speeding up or slowing down A derivative is just: How fast are you going RIGHT NOW? Not how far did you swing? That's distance. Not how long did it take? That's time. Derivative equals change in distance divided by change in time. That's it. Everything else is just fancier ways of asking how fast are things changing? Want to see how this connects to the scary math symbols?

Yes! But I don't understand why we need fancy notation for how fast are things changing.

Great question! Here's why: Speed equals 100 mph tells you how fast you're going ON AVERAGE over an hour. But what if you're at a stop sign, then floor it? Your speed is constantly CHANGING. The derivative tells you: What's your speed EXACTLY NOW? The notation: dy/dx means how y changes as x changes. F prime of x means the derivative (rate of change) of f at point x. These are just fancy ways of saying instantaneous speed. The magic: Calculus lets us capture CHANGE at a single instant, not just average change over time. Want to see how we actually calculate this?

But how can you have change at a single instant? If time is frozen, there's no change!

THIS IS THE BRILLIANT INSIGHT THAT INVENTED CALCULUS! You're absolutely right: At a frozen instant, there's no change. So how do we define speed right now? The trick: We zoom in closer and closer and closer. 1. What's your average speed over 1 hour? 2. Over 1 minute? 3. Over 1 second? 4. Over 0.000001 seconds? As the time interval gets TINY, the average speed approaches the instantaneous speed. Mathematically: average_speed = distance divided by time. Instantaneous_speed = limit as time approaches 0 of distance divided by time. We're asking: What does the average speed approach as the time interval gets infinitely small? The answer is the derivative. This insight (independently discovered by Newton and Leibniz) solved the change at an instant paradox and launched modern science. Want to see how we actually compute this limit?