system
Given the following post, respond as if you're commenting on Reddit:
human
ELI5: Calculus Derivatives

I'd like to know as much as possible about derivatives; from how they are solved to what are their applications. Thx in advance.
gpt
If you actually want to know *as much as possible* about derivatives, you're in the wrong place. But I'll explain the concept LY5. (and let someone else do the math part if they want.)

A derivative tells you how fast something is changing. It's a number telling you a *rate of change*.

Let's say we're driving down the road in a car. At 5 o'clock we are at mile marker 10, and at 5:01 we are at mile marker 11. Our position on the road was 10 at time 5:00, and it was 11 at time 5:01.  So how much did our position change? It changed by one mile in one minute. So, we were moving at 60 miles per hour.

That's why we say that velocity (or speed in a car) is the *derivative* of position. If you took a graph with position on the y-axis and velocity on the x-axis, you would see that, at time 5:00, we were at position 10, and at time 5:01 (one unit to the right), we were at position 11 (one unit upward). So if we drew a line from the first point to the second, it would look like this:
    
    12 |
       |
       |
    11 |         o
       |        /
       |      /
       |    /
    10 |  o
         -------------------------
         5:00   5:01   5:02   ...

This line tells us our **position** over time. At 5:00, we were at marker 10; at 5:01, we were at marker 11, and so on.

And so, we can ask, what's the *slope* of that line? It goes up by one unit and over by one unit. So for every one (mile) it goes up, it goes one (minute) to the right. So the slope is **1 mile/minute**, which is the derivative of our position.

Here's some extra food for thought:  When you step on the gas pedal of your car, the car speeds up, right?  So say you're going 60 mph when you hit the gas, hard, and in two seconds, you've gone up to 70 mph.  So in those two seconds, your *velocity* has changed by 10 miles per hour. So what is the rate of change of your velocity? It changed by 5 miles per hour each second. (It did this for the two seconds you hit the gas, making a total change of 10 miles per hour -- from 60 up to 70.)  So your *acceleration* -- the derivative of velocity -- was: (5 miles per hour) per second. Just like velocity is the derivative of position, acceleration is the derivative of velocity.