ctrawyfbil
05–29–25
calculate the rate at which you’re falling behind in life
derivatives for dummies
I learned derivatives at the ripe age of 22 years old… which is approximately four years later than most U.S. students will master the concept.
Why did this happen? Well, it was mostly a culmination of poor high school math requisites and a couple of gap years - excuses, excuses… there I was anyway, squeezing out tears of frustration behind gritted teeth, fretting over my derivatives all the while holding tight to my high school valedictorian superiority complex (I try to only let myself access it when I’m feeling truly pathetic, like losing a chess game or, of course, doing math homework).
But the entire time I couldn’t help but feel like the practice was some kind of sick joke. The concept itself is so paradoxical.
Supposedly, a derivative represents an “instantaneous rate of change,” but in the same breath we’re taught how to calculate it, we’re also told that it doesn’t really exist. There is no change in a single instant, because an instant doesn’t exist. Time isn’t paused. Motion always occurs over an interval, no matter how small.
So what we’re really calculating isn’t motion in a true “instant,” but the average rate of change between two points in time that are very, very close together.
“Okay, Brinlee, you genius, that’s the whole point of derivatives.”
Yeah, yeah, but the formula isn’t the part that truly irked me.
It was the fact that every problem featuring some kind of racing car or a flying rocket ship had the derivative equal to zero when the object was zero seconds into its journey.
And like, okay - when the journey begins, the car is sitting still. Makes sense, right? But actually… no it doesn’t???
If the car isn’t moving at the first moment it begins to move, then when does it start moving? How can the derivative at time = 0 be equal to 0? When does motion begin, if not at 0 seconds?
0.000000001 seconds?
0.00000000000000000001 seconds?
0.0000000000000000000000000000001 seconds?
But not at true 0. The derivative of any object at the very start of its journey is 0.
And before you second-guess me based on my background as a 22-year-old doing Calc 2 homework, just know that I looked it up and it really is a paradoxical inaccuracy.
But as much as I am a devout hater of all derivatives, I can’t help but feel like the beginnings of everything fall in line with this very same paradox.
Lately, I’ve been feeling like I’m constantly at the start of things, waking up each day feeling like life is a race I keep showing up late to. Feet heavy like cement, staring down a never-ending journey dotted by someone else’s perfect acceleration curve. It’s not that I don’t feel like I’ve done enough… I feel like I’ve never done anything at all. I often question why I even let myself dream when I’m still at the very start, seemingly standing completely still.
But really, my mindset is an issue of measurement. The way I evaluate “progress” doesn’t capture what’s actually happening. From a distance, when comparing this one moment in time to the potential of my entire life plan, my average rate of change may appear to be zero. But zoom in, and the story changes.
The derivative of a moving object at 0 seconds can’t actually be 0 - that’s just the simplified answer. In reality, the velocity is probably something more like 0.00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001
Which is not nothing! Even if it is almost nothing hehe
The car is still moving. The rocket is still moving. I’m still moving.
For this reason, I don’t think anyone should ever really be embarrassed of their journey, because there’s no single formula for the way each of us moves through life. If you take the derivative of a single instant in time across two people’s graphs, even those that ultimately ended up in the same place, the values will always be different... and relatively innacurate.
And when you’re at the beginning of your journey, or even stuck in a messy middle, it can seem as if you're not moving at all. But you are. Quietly. Invisibly. Undeniably.
There’s no way to exactly calculate your progress in a single moment…
unless you stop.
Why did this happen? Well, it was mostly a culmination of poor high school math requisites and a couple of gap years - excuses, excuses… there I was anyway, squeezing out tears of frustration behind gritted teeth, fretting over my derivatives all the while holding tight to my high school valedictorian superiority complex (I try to only let myself access it when I’m feeling truly pathetic, like losing a chess game or, of course, doing math homework).
But the entire time I couldn’t help but feel like the practice was some kind of sick joke. The concept itself is so paradoxical.
Supposedly, a derivative represents an “instantaneous rate of change,” but in the same breath we’re taught how to calculate it, we’re also told that it doesn’t really exist. There is no change in a single instant, because an instant doesn’t exist. Time isn’t paused. Motion always occurs over an interval, no matter how small.
So what we’re really calculating isn’t motion in a true “instant,” but the average rate of change between two points in time that are very, very close together.
“Okay, Brinlee, you genius, that’s the whole point of derivatives.”
Yeah, yeah, but the formula isn’t the part that truly irked me.
It was the fact that every problem featuring some kind of racing car or a flying rocket ship had the derivative equal to zero when the object was zero seconds into its journey.
And like, okay - when the journey begins, the car is sitting still. Makes sense, right? But actually… no it doesn’t???
If the car isn’t moving at the first moment it begins to move, then when does it start moving? How can the derivative at time = 0 be equal to 0? When does motion begin, if not at 0 seconds?
0.000000001 seconds?
0.00000000000000000001 seconds?
0.0000000000000000000000000000001 seconds?
But not at true 0. The derivative of any object at the very start of its journey is 0.
And before you second-guess me based on my background as a 22-year-old doing Calc 2 homework, just know that I looked it up and it really is a paradoxical inaccuracy.
But as much as I am a devout hater of all derivatives, I can’t help but feel like the beginnings of everything fall in line with this very same paradox.
Lately, I’ve been feeling like I’m constantly at the start of things, waking up each day feeling like life is a race I keep showing up late to. Feet heavy like cement, staring down a never-ending journey dotted by someone else’s perfect acceleration curve. It’s not that I don’t feel like I’ve done enough… I feel like I’ve never done anything at all. I often question why I even let myself dream when I’m still at the very start, seemingly standing completely still.
But really, my mindset is an issue of measurement. The way I evaluate “progress” doesn’t capture what’s actually happening. From a distance, when comparing this one moment in time to the potential of my entire life plan, my average rate of change may appear to be zero. But zoom in, and the story changes.
The derivative of a moving object at 0 seconds can’t actually be 0 - that’s just the simplified answer. In reality, the velocity is probably something more like 0.00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001
Which is not nothing! Even if it is almost nothing hehe
The car is still moving. The rocket is still moving. I’m still moving.
For this reason, I don’t think anyone should ever really be embarrassed of their journey, because there’s no single formula for the way each of us moves through life. If you take the derivative of a single instant in time across two people’s graphs, even those that ultimately ended up in the same place, the values will always be different... and relatively innacurate.
And when you’re at the beginning of your journey, or even stuck in a messy middle, it can seem as if you're not moving at all. But you are. Quietly. Invisibly. Undeniably.
There’s no way to exactly calculate your progress in a single moment…
unless you stop.
so just don’t stop :)