# What does the derivative really calculate?

#### chesterfiuk

##### New member
I learned several derivation rules and everything in the mechanical engineering course, and now, I'm going to calculus II, but I still have my doubts about what the derivative calculates. Some videos on youtube say that it calculates an INSTANT VARIATION at a point, but others convey the idea that the derivative actually calculates a rate of change in a very small region around the point (since an instant variation at a point it would be impossible since we need 2 distinct moments whose difference between them would result in a variation) which when it is increased, has the same slope as the tangent line.

But, who to trust?

In college they didn't explain exactly what she calculates. I would like an answer that defines what the derivative really calculates, just making calculations without really knowing what I'm really calculating, makes my desire to continue the course less and less

And, I'm Brazilian, so, sorry in advance for possible typos, thank you in advance

#### Subhotosh Khan

##### Super Moderator
Staff member
I learned several derivation rules and everything in the mechanical engineering course, and now, I'm going to calculus II, but I still have my doubts about what the derivative calculates. Some videos on youtube say that it calculates an INSTANT VARIATION at a point, but others convey the idea that the derivative actually calculates a rate of change in a very small region around the point (since an instant variation at a point it would be impossible since we need 2 distinct moments whose difference between them would result in a variation) which when it is increased, has the same slope as the tangent line.

But, who to trust?

In college they didn't explain exactly what she calculates. I would like an answer that defines what the derivative really calculates, just making calculations without really knowing what I'm really calculating, makes my desire to continue the course less and less

And, I'm Brazilian, so, sorry in advance for possible typos, thank you in advance

https://www.quora.com/What-are-the-applications-of-derivatives-in-mechanical-engineering

https://www.quora.com/What-are-the-application-of-derivatives-in-engineering

#### Dr.Peterson

##### Elite Member
In college they didn't explain exactly what she calculates. I would like an answer that defines what the derivative really calculates, just making calculations without really knowing what I'm really calculating, makes my desire to continue the course less and less

The easiest way to see what it does may be to think of the graph of a function. Pick a point on the graph, and draw a line tangent to it. The derivative at that value of x gives the slope of the tangent line you drew.

Try doing this on the actual graph of a specific function, such as y = x^2, to see it most clearly.

After you've done that, you can think about what you've got. The tangent line is only actually equal to the function at the one point; but very close to the point, the slope from the chosen point to another nearby will be very close to the slope of the tangent line -- the closer the point, the closer the slope.

The derivative is the limit of the slope of such "secant" lines, as the second point moves toward your fixed point. In this sense, both answers you found are correct. The derivative gives the "instantaneous" rate of change, in the sense of the slope right at that point; but since as you say that doesn't quite make sense, that is defined as the limit of rates of change over smaller and smaller intervals near the point.

By the way, you are not alone. When calculus was first invented, it was explained in terms of fictional quantities that were imagined to be both zero and non-zero at the same time, which left a lot of people unsure of the validity of calculus. It was only in the 1800's, with the development of the theory of limits, that calculus became proper mathematics that could be proved.

#### Jomo

##### Elite Member
It is sad that your teacher never explained what a derivative is. It is good that this bothers you.

Here is a good video describing what a video does.

#### chesterfiuk

##### New member
It is sad that your teacher never explained what a derivative is. It is good that this bothers you.

Here is a good video describing what a video does.

It is sad that your teacher never explained what a derivative is. It is good that this bothers you.

Here is a good video describing what a video does.

Thanks for the video and help, and it's really sad. I see my colleagues simply doing the calculations without understanding the real meaning and thinking that they understand everything.

For some people, what matters is not really learning, but just taking an average score and passing, memorizing questions or cheating, I see so many people who are deluded by just one number at the end of the semester without worrying whether they really learned or not , this limits the person and prevents him from learning more.

#### chesterfiuk

##### New member
The easiest way to see what it does may be to think of the graph of a function. Pick a point on the graph, and draw a line tangent to it. The derivative at that value of x gives the slope of the tangent line you drew.

Try doing this on the actual graph of a specific function, such as y = x^2, to see it most clearly.

After you've done that, you can think about what you've got. The tangent line is only actually equal to the function at the one point; but very close to the point, the slope from the chosen point to another nearby will be very close to the slope of the tangent line -- the closer the point, the closer the slope.

The derivative is the limit of the slope of such "secant" lines, as the second point moves toward your fixed point. In this sense, both answers you found are correct. The derivative gives the "instantaneous" rate of change, in the sense of the slope right at that point; but since as you say that doesn't quite make sense, that is defined as the limit of rates of change over smaller and smaller intervals near the point.

By the way, you are not alone. When calculus was first invented, it was explained in terms of fictional quantities that were imagined to be both zero and non-zero at the same time, which left a lot of people unsure of the validity of calculus. It was only in the 1800's, with the development of the theory of limits, that calculus became proper mathematics that could be proved.

My God, your answer helped me a lot, thank you very much.

And as you said, it really helps to draw the graph and move the interval between the 2 points closer to 0, you can clearly see that it really gets closer and closer to the tangent line. The idea of getting closer to 0 but not reaching 0 in order to maintain a variation even if it is very small is very bright.

So, these small variations around point X for example, having point X fixed, the smaller the variations, the closer they will come to that value that is the derivative. I now also notice that, although the term instantaneous variation in a point does not make sense, in a way the term is not completely wrong, since there are so small variations around the point that it could be considered practically an instant for us.

The derivative is the most accurate approximation that we can have for these small variations around the point, since they will always approach but still will not assume this value, since for that, this variation would have to be 0, and in that case there would be no variation.

It is also interesting that this idea of using these increasingly smaller approximations does not only work for the derivative, from my point of view as a student, this is the idea that supports the integral as well, it is the basis of the calculation.

#### pka

##### Elite Member
By the way, you are not alone. When calculus was first invented, it was explained in terms of fictional quantities that were imagined to be both zero and non-zero at the same time, which left a lot of people unsure of the validity of calculus. It was only in the 1800's, with the development of the theory of limits, that calculus became proper mathematics that could be proved.
To say “ explained in terms of fictional quantities” is a bit harsh for a modern mathematician. It is true that Bishop George Berkeley, a contemporary of Newton, wrote a devastating critique of the logic of Newton's infinitesimals. He call them ghosts of dearly departed numbers.
But as it was with so-called imaginary numbers we found it possible to enlarge the real numbers with the addition of a single symbol $$\bf{i}$$ that is the solution to $$x^2+1=0$$ So was it for Abraham Robinson in the 1950's to work out the logical basis for infinitesimals . He worked out the basis if a new field of Non-Standard Analysis. There is available a free download of a calculus textbook: Elementary Calculus: An Infinitesimal Approach by Jerome Keisler. The chapters and whole book is a free down-load at It's FREE and complete.

#### Jomo

##### Elite Member
Thanks for the video and help, and it's really sad. I see my colleagues simply doing the calculations without understanding the real meaning and thinking that they understand everything.

For some people, what matters is not really learning, but just taking an average score and passing, memorizing questions or cheating, I see so many people who are deluded by just one number at the end of the semester without worrying whether they really learned or not , this limits the person and prevents him from learning more.
The video I posted sounds like it shows exactly what you said Dr Peterson told you. That is the only way in my opinion to show the meaning of a derivative.

#### HallsofIvy

##### Elite Member
I learned several derivation rules and everything in the mechanical engineering course, and now, I'm going to calculus II, but I still have my doubts about what the derivative calculates. Some videos on youtube say that it calculates an INSTANT VARIATION at a point
YES!
Or, more correctly, rate of change, not amount of variation.

but others convey the idea that the derivative actually calculates a rate of change in a very small region around the point (since an instant variation at a point it would be impossible since we need 2 distinct moments whose difference between them would result in a variation)
NO!
That is not how the derivative is defined. "2 distinct points" give a average rate of change between those two points. The derivative requires the limit that as the distance between the two points. Didn't your Calculus I textbook, if not you teacher, emphasize the limit concept? That is the crucial idea that defines "Calculus" and makes it different from "Algebra"!

which when it is increased, has the same slope as the tangent line.

But, who to trust?

In college they didn't explain exactly what she calculates.
That's odd. I don't know who you mean by "they" but every textbook I have seen has made it clear that the "derivative" is the "instantaneous rate of change" or "slope of the tangent line". In fact, you just used the phrase "slope of the tangent line" yourself, above. Aren't you clear on what that means? Do you know what the "slope of a line" tells you? Do you know what the tangent line to a curve, at a point, is?

I would like an answer that defines what the derivative really calculates, just making calculations without really knowing what I'm really calculating, makes my desire to continue the course less and less
If the definitions in Calculus texts aren't good enough, I am not sure what kind of definition you want. The derivative of a function tells you the instantaneous rate of change. If you know that some object has moved from x= 0, at time t= 0, to x= 10, at time t= 1, then you know that the average speed (rate of change of position) is (10- 0)/(1- 0)= 10. If you know that the "position function" is x= 10t^2, then you know that, at each time t, between 0 and 1, the instantaneous speed, the speed at that instant, not over some span of time, was dx/dt= 20t.
And, I'm Brazilian, so, sorry in advance for possible typos, thank you in advance
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#### chesterfiuk

##### New member
YES!
Or, more correctly, rate of change, not amount of variation.

NO!
That is not how the derivative is defined. "2 distinct points" give a average rate of change between those two points. The derivative requires the limit that as the distance between the two points. Didn't your Calculus I textbook, if not you teacher, emphasize the limit concept? That is the crucial idea that defines "Calculus" and makes it different from "Algebra"!

That's odd. I don't know who you mean by "they" but every textbook I have seen has made it clear that the "derivative" is the "instantaneous rate of change" or "slope of the tangent line". In fact, you just used the phrase "slope of the tangent line" yourself, above. Aren't you clear on what that means? Do you know what the "slope of a line" tells you? Do you know what the tangent line to a curve, at a point, is?

If the definitions in Calculus texts aren't good enough, I am not sure what kind of definition you want. The derivative of a function tells you the instantaneous rate of change. If you know that some object has moved from x= 0, at time t= 0, to x= 10, at time t= 1, then you know that the average speed (rate of change of position) is (10- 0)/(1- 0)= 10. If you know that the "position function" is x= 10t^2, then you know that, at each time t, between 0 and 1, the instantaneous speed, the speed at that instant, not over some span of time, was dx/dt= 20t.
And, I'm Brazilian, so, sorry in advance for possible typos, thank you in advance
[/QUOTE]

So tell me, how can a point have instant variation? If variation is just something that occurs between 2 points?

#### tkhunny

##### Moderator
Staff member
So tell me, how can a point have instant variation? If variation is just something that occurs between 2 points?
I'll ignore the first question.
For the second: No. This is what the limit does for us.

1) Do you have an automobile?
2) Does it have a speedometer?
3) What does the speed on that instrument mean? How well can you explain it? When you look at it and read a number, what information are you gaining?

#### HallsofIvy

##### Elite Member
The LIMIT concept! The derivative is the [b\]limit[/tex], as h goes to 0, of the "difference quotient", $$\displaystyle \frac{f(x_0+ h)- f(x_0)}{h}$$. The difference quotient is between two different points $$\displaystyle (x_0, f(x_0))$$ and $$\displaystyle (x_0+h, f(x_0+ h)$$ but the derivative, the limit as h goes to 0, depends only on $$\displaystyle x_0$$.

#### chesterfiuk

##### New member
My point is that there is no instantaneous variation (it is a wrong term), variation is something that occurs between 2 instants. If I show you 1 picture of a car moving on a track, and then another picture showing another car on the same track, and that's it, would you be able to tell me which one is faster, or physically speaking, which has the most variation ?

#### Jomo

##### Elite Member
Fine. Can you please give us a better term to use?

#### HallsofIvy

##### Elite Member
No, you wouldn't. You can have an "instantaneous variation" or, if you don't like that term, "instantaneous velocity". You have to use the limit

#### Dr.Peterson

##### Elite Member
My point is that there is no instantaneous variation (it is a wrong term), variation is something that occurs between 2 instants. If I show you 1 picture of a car moving on a track, and then another picture showing another car on the same track, and that's it, would you be able to tell me which one is faster, or physically speaking, which has the most variation ?

But I think we all agree on that; only you have used the word "variation" (taken from a video you saw?) In post #10, that was corrected to "rate of change".

You're right that "instantaneous variation" makes no sense. But "instantaneous rate of change" does, and that is what we have all been talking about.

#### chesterfiuk

##### New member
Fine. Can you please give us a better term to use?
" that is defined as the limit of rates of change over smaller and smaller intervals near the point "

#### tkhunny

##### Moderator
Staff member
" that is defined as the limit of rates of change over smaller and smaller intervals near the point "

The problem you are having is stopping at some finite place. When it says "smaller and smaller" it does not say to stop doing that. Keep doing it. If you stop, you are not understanding the limit. This is how ancient thinkers used to do it. They believed if you chopped things into small enough pieces you could explain things better. This isn't totally false, but the limit rejects this idea. Keep chopping until your pieces are size zero (0).

We do not use the language "The limit approaches...". (Granted, many a student has made this mistake.)
We do use the language "The limit is...".

An obvious question is this: Is 1 = 0.9999...? Does 0.999... approach 1 or is it 1? What say you?

#### HallsofIvy

##### Elite Member
My point is that there is no instantaneous variation (it is a wrong term), variation is something that occurs between 2 instants. If I show you 1 picture of a car moving on a track, and then another picture showing another car on the same track, and that's it, would you be able to tell me which one is faster, or physically speaking, which has the most variation ?
Yes, all of that is true and is the heart of the "derivative"! You started by asking about the derivative but you are refusing to listen!

Actually the point you make here was one reason why the "Calculus" was invented in the first place. Newton had determined that the gravitational force on the moon or on a planet depended on the distance from the attracting body (1 over the square of the distance). But that does have a specific distance at a single specific time so must have a specific acceleration (and so a specific speed) at that time! Newton and Leibniz developed the Calculus specifically because the had to overcome your objection!