What would happen if I took the derivative of a function at a certain point?

B-but it's supposed to ramp up so each subsequent generation becomes greater until your family become literal gods, meanwhile great great grandfather (you) is looking up from the heavens and wondering about what you want to reincarnate into next

you find how much one variable fluctuates in relation to another.

For example, let's say you have a function of position vs.time. Let's also pretend the relationship between position and time is not super obvious, and the relation is "physical" (i.e. is able to be differentiated).

The derivative of position vs. time at a specific time, "t", is the instantaneous variation of position with respect to time AT the time t. In this case, the time-derivative of position at time t is just the velocity at t.

Pretend you are at a stoplight and you hit the gas, accelerating forward. If you drew a position vs time graph, you would see that as time progressed, the distance you drove would be greater and greater -- in fact, the longer you drove, the more the position would change per second.

As you accelerate, you look at the speedo and see your car's speed is increasing. At time t, you see your car is moving at 55mph. So at time t, the derivative of position with respect to time is 55mph AT THAT INSTANT. One second before t, maybe your car is going 45 mph. One second after t, maybe your car is going 60mph. But by taking the derivative at t, you find the INSTANTANEOUS speed at t. Notice that the instantaneous velocity is not simply (60-54)/(2 seconds), but it is a snapshot of an infinitesimally small period of time.

Another way to think of the problem: let's say you have a special instrument that measures the position and time of the car. So for example, you use the instrument once to measure p_1 and t_1 (the car's position at time t_1). You then use the instrument again and measure (p_2,t_2). You can measure the speed of your car by taking two measurements and using v= (p_2 - p_1)/(t_2 - t_2).

This is fine if you want an average speed over a time period between t_1 and t_2. If you travel 40 miles in 1 hour, your average speed was 40mph. But what if you want to know your exact speed at t_1?

Well, you can make the interval smaller. if you travel 0.75 miles in one minute, you had a speed of 45 mph within that 1 minute interval. Want more precise? Measure the positions within an interval of 1 sec... 0.013 miles travelled withing 1 sec -> 47mph -- an even better estimate for the instantaneous speed.

Basically, the idea of a time-derivative is that you reduce the interval of time to zero, so you are quite literally finding the speed at one single instant. In a way, you are finding the change of position for the car in an interval that has no duration.

In the example above, we took the slopes of lines, but each time we made the distance between points smaller and smaller. Eventually, we ended up with the slope of the line at a single point in the position vs time function.

Sorry if this is a bad explanation, it was off-the-cuff.

kek. you're supposed to shitpost not give him an actual mini physics lesson lel

>Update
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>Lel bruh just shitpost kekimus
i truly get the [math]\lim{x\rightarrow \infty} Veeky Forums = /b/[/math] memes now

It would be extremly painful

You mean a dot over the function.

>gets buttmad because I don't do what you expect

If Veeky Forums is for shitposting, and I do the opposite, am I not shitposting as well?

...

lotta loyalty for a hired assistant

The d is supposed to be in upright position.

Thank you for actually trying to be helpful on Veeky Forums

here have a rare pepe