Derivatives

[KvickFlygarn87]

Can anyone explain them intuitively? I'm an 8th grader and I excel (dat humble tho) in maths. I recently wanted to get into calculus, so I figured derivatives is the place to start. My teacher's math book was not much help (it was almost exclusively work tasks, not any explanations) so now I'm asking you. What are they? At first I thought that they were the rate at which y changed in an equation, but internet tells me no. Also, how do I calculate them? Somewhere I found something like

where h is the distance of a horizontal line from x.

if I recall correctly. Could someone give me a deep explanation of derivatives?

[stupid_chris]

Can anyone explain them intuitively? I'm an 8th grader and I excel (dat humble tho) in maths. I recently wanted to get into calculus, so I figured derivatives is the place to start. My teacher's math book was not much help (it was almost exclusively work tasks, not any explanations) so now I'm asking you. What are they? At first I thought that they were the rate at which y changed in an equation, but internet tells me no. Also, how do I calculate them? Somewhere I found something like

http://latex.codecogs.com/png.latex?\lim_{h\rightarrow&space;0}&space;\frac{f(x+h)&space;-&space;f(x)}{h}

where h is the distance of a horizontal line from x.

if I recall correctly. Could someone give me a deep explanation of derivatives?

Basically, mathematically speaking, a derivative is the slope of the tangent at one precise point in a curve. So and instantaneous variation rate.

I don't have much time to explain as of right now as I need to go to class, but I reccommend you first start looking into limits. If you don't understand limits, derivatives will be much harder to grasp.

Probably other people will answer, but I can guide you around later on #kspofficial once I get home if you want to

[Sillychris]

You've got it right, buddy. A derivative describes how quickly the y changes with respect to variations in x.

Example: How fast does your position change wrt time? This is velocity. (dx/dt=v)

How fast does your velocity change wrt time? This is acceleration. (dv/dt=a)

[Red Iron Crown]

A derivative is a rate of change. It's really as simple as that.

stupid_chris' definition as the slope of a curve at a point is correct, too, but doesn't give an intuitive grasp of what they are for. He is also correct in that you must understand how limits work before getting into derivatives or you'll find them very difficult.

Some examples would be: The derivative of distance with respect to time is speed. The derivative of speed with respect to time is acceleration. The derivative of fuel amount with respect to distance is fuel consumption (in the L/100km or mpg sense).

[KvickFlygarn87]

I sorta understand limits. Like, if lim x->0, that means x can approach 0 but never turn to 0.

[Red Iron Crown]

I sorta understand limits. Like, if lim x->0, that means x can approach 0 but never turn to 0.

If you look at the limit you posted in the OP, you should be able to see that it is calculating slope, rise over run. The limit is reducing the amount of run to an infinitesimal amount. But not zero, because then you'd be dividing by zero. This is useful for getting the slope of a curve rather than a line, where you can just take two points and calculate slope.

[Ravenchant]

http://latex.codecogs.com/png.latex?\lim_{h\rightarrow&space;0}&space;\frac{f(x+h)&space;-&space;f(x)}{h}

That limit is usually used for defining a derivative, yes. As x approaches zero, you can think of the path of the function being split into shorter and shorter straight lines, which are tangent to the path. Now, the finer the division, the better those lines approximate the function path. X -> 0 simply means that the divisions are becoming infinitely small, and thus the "lines" exactly follow the curve, e.g. there is no approximation anymore. Then you can start talking about the slope of the function in any given point- it can be given as f', or df/dx, or a dot over f. The first derivative is kinda the "rate of change" of the curve, then you can take the derivative from that and get the rate of change of the slope, etc.

[PlonioFludrasco]

For example,average speed is calculated as an interval of space divided by an interval of time:

V=(deltaX)/(deltaT)

If you want to calculate instantaneous speed you want that your interval of time is zero. But you cannot divide by zero, and here comes the limit: you want a very little deltaT, but it can't be zero, so you take an infinitesimal (=as little as you want) interval of time, called dT.

V(T) = lim X(T+dT) - X(T)/dT

Where dT is something very little: dT->0

Change the name of the variables and you have the formal definition of the derivative.

PS Usually, you won't calculate a derivative with the definition formula, because it's long and boring, there are some much simple formulas and theorems that tells you how the derivative of a certain kind of function is made.

For example, for functions like x^n the derivative is nx^(n-1).

[stupid_chris]

I sorta understand limits. Like, if lim x->0, that means x can approach 0 but never turn to 0.

Understanding them is a thing. Knowing how to use them or calculate them is another.

For example, you already understand what a derivative is now: it's a variation rate on an infinitely small interval But if you want to get to calculate derivatives, you'd ought to understand how to calculate limits, and how to solve 0/0, k/0, ∞/∞, ∞-∞ and 0*∞ cases

When working with derivatives, you will rarely use that limit. This is the definition of it, but there are formulas to calculate derivatives of specific functions, as well as multiplication/division/addition of derivatives. They all sort of "flow" from there, but when you start doing calculus you very quickly leave this formula behind.

[Linear]

...is the rate of change.

Differentiation is a simple process where you times by the power and minus one to the power, i.e f(x) = x^2, f'(x) = 2x where f'(x) is the first derivative of the function (x).

Simple, yet golden. It's basically working out the gradient if you want a simpler view of it.

Integration is the 'opposite'. Add one to power, divide by new power =]

[stupid_chris]

That is for x functions. The derivative of Arctan(x²) is a whole other deal

[Chrisd857]

That is for x functions. The derivative of Arctan(x²) is a whole other deal

The integral is worse. At least it's elementary.

[K^2]

That is for x functions. The derivative of Arctan(x²) is a whole other deal

You can still work it out using (almost) the same information. You need series expansion of sin and cos functions and the chain and product rules.

tan(x) = sin(x) / cos(x)

sin(x) = x - x³/3! + x⁵/5! - ... [These are series expansions. Technically, they are derived using derivatives, so it's cheating a little... This can be derived differently, but it will take much longer.]

cos(x) = 1 - x²/2! + x⁴/4! - ...

d sin(x)/dx = 1 - x²/2! + x⁴/4! - ... = cos(x) [Taking derivatives using rules from last page]

d cos(x)/dx = ... = -sin(x)

tan(x) = sin(x) cos^-1(x)

d tan(x)/dx = d sin(x)/dx cos^-1(x) - sin(x) cos^-2(x) d cos(x)/dx = 1/cos²(x) [using chain rule and product rule.]

x = arctan(tan(x)) [definition]

dx/dx = d arctan(tan(x))/dx

1 = d arctan(y)/dy d tan(x)/dx taken at y = tan(x) [Chain rule, again.]

1 = d arctan(y)/dy / cos²(x)

d arctan(y)/dy = cos²(x)

Here I have to use a bit of trickery. Lets call cos(x) = z. In that case, sin(x) = Sqrt(1-z²).

d arctan(y)/dy = z² taken at y = Sqrt(1-z²)/z

We can solve for z now,

y² = (1-z²)/z²

z²y² = 1 - z²

z² = 1/(1+y²)

And so we have our derivative of arctangent.

d arctan(y)/dy = 1/(1+y²)

And finally, using chain rule one last time, we get the required.

d arctan(x²)/dx = 2x/(1+x⁴)

Naturally, it's way easier if you know some of the shortcuts. But the point is that just knowing the rule for xⁿ and the chain/product rules, you can work out almost everything.

Integrals are a different story.

Edited April 1, 2014 by K^2

[TheGatesofLogic]

Yeah, many integrals are not quite so easily formulated from scratch and when they start getting complex knowing how to solve by parts becomes utter necessity, and that isn't something one can do with as comparably small a methodology as derivatives require.

BTW when I was in eighth grade my grandfather gave me a very old copy of Philosophæ Naturalis Principia Mathematica and I fell in love with the book. I highly recommend it for your beginnings in calculus as it tends to describe things in a way seldom used in more formal calculus courses and leads to an excellent basis in ones ability to understand the reasons for the equivalence between the different methods of integration and derivation. Although its heavy in classical mechanics (I assume this will be a non-issue as it was for me since you are taking calculus at such an age, ability to understand mechanics and ability to understand algebra/trigonometry/calculus tend to go hand in hand) it is also very math oriented and in my opinion is a must-read for anyone even considering taking any calculus course. Although it might seem like an old and boring book, Newton was excellent at describing his methodology within it and it is far more enjoyable than the modern textbooks one sees on a daily basis.

Edited April 2, 2014 by TheGatesofLogic

[K^2]

It's not bad, but a lot of its perspectives are dated. I think, a modern introductory Real Analysis book is a better starting point. Rudin's Real and Complex Analysis book is fantastic, but it's going to be very heavy for someone who isn't used to formal Mathematics. It's also very difficult to follow if you don't know what Fields and Topological Spaces are. For someone who just wants to understand theory behind Calculus a little better, Introductory Analysis: The Theory of Calculus by Fridy is really not bad. It starts with fairly straight forward concepts, doesn't get too abstract, and has many good examples. At the same time, it covers all the bases in terms of limits, differentiation, and integrals in Riemann Integral sense. In short, it teaches you all the Analysis you can learn without having to understand Abstract Algebra, Topology, and Measure Theory.

Edited April 2, 2014 by K^2

[TheGatesofLogic]

Well certainly it requires accompaniment by a more comprehensive and modern text, I recommended primarily because it helps (well helped me at least) some people understand the equivalence between different (and more often than not more elegant; Newton's personal notation is not something I would recommend compared to the notation used by Leibniz) approaches to the same problems when read in such conjunction with some other work, which I believe is something too often ignored in modern math classes. It's also fascinating from a historical perspective, however that's a little bit off of the topic, and more of personal opinion than anything else.

[K^2]

Yeah, it's definitely good for historical context. I just feel like it can sort of give you some wrong ideas if you start with it. I'd recommend learning some basics of Analysis before reading Principia. Just so that you have a better idea of what's on the right track and what isn't.

[KvickFlygarn87]

Glorious! You guys are the greatest, you know that? You made my evening! I had half expected it to have one page tops, and no in-depth answers - I need to get off Battlelog forums - but you guys did it once again! I'll look into those books. They don't happen to be availaible in eBook/pdf form do they?

[TheGatesofLogic]

I'm not sure it would be easy to find Newton's Principia on a kindle or nook if that's what you mean, but since it has been out of copyright protection for just under 240 years (actually it was published a hundred years before international copyright law was even established) it is now freely available on the internet in several places, and a quick search netted me this site https://archive.org/details/newtonspmathema00newtrich which supplies an variety of different file formats, but unfortunately sports a long preface on newton's life at the beginning that I advise you simply skip over. I'm not quite sure on the book k^2 mentioned and since i am not entirely familiar with it, it may still be in copyright, in which case you would have to buy it.

[K^2]

Yes, Fridy's book is certainly under copyright. You can get it on Amazon. I don't know if they offer a digital version, though.

[Ravenchant]

I have one really great advanced mathematics (actually Analysis I) book I don't need anymore and could mail away, the only downside is that it's in Slovenian

There surely are some copyright-free scripts and textbooks to be found online, and you can try searching for lecture videos...like this one (it's probably a bit too in-depth for 8^th grade, but still)

[Nascosto]

The Formula you linked actually has a little bit more to it in order to understand it intuitively. You know it better as (Y2-Y1)/(X2-X1). It's literally the slope formula. However, instead of Y1, we use "f(x)". Instead of Y2, we use "f(x+h)" where h represents how far away x-wise the second point is from the first. For X1 we just use X, and for X2 we use "x+h". The thing is, the top stays exactly like you'd expect. The bottom however simplifies pretty significantly, because when you subtract x from (x+h), you just get h. Calculus, at least Derivatives, revolve around what happens when h gets very very small, or when the two points you're finding the slope between get very close together. The closer they get, the less the slope changes, and by looking at those patterns we can see what the slope is heading towards if those two points were to intersect. This is terrible terrible official form and people will go all ape**** to hear me say it, but in layman's terms, Derivatives represent the instantaneous slope of a curve at a given point. - your friendly, lowly, high school math teacher.

[K^2]

The closer they get, the less the slope changes

It is probably worth noting that this is not a guarantee, but it is a requirement for derivative to be defined. A good counterexample is f(x)=x sin(1/x) for x not equal to 0 and f(0)=0. This is a continuous function defined for all real numbers. However, it does not have derivative at 0, precisely, because the slope changes more and more as h gets small.

[KvickFlygarn87]

@Ravenchant Aw, too bad. I think learning Slovenian would be a taking it a tad too far I'll take a peek at that video Soon^TM!

On an unrelated note, my family has a couple of friends down in Slovenia that we go visit on the summer every year.

@Nascosto Great explanation, I think I just got a tad smarter

@K^2 Alright! I'll keep that in mind, derivatives of sin (and probably other trig functions too) are weird.

[Doozler]

I'll keep that in mind, derivatives of sin (and probably other trig functions too) are weird.

K^2 said "sin(1/x)"! Actually, Sin (x) and cos(x) are fine. In fact, here's your first assignment (should you want one! ):

• Plot a graph of Cos(x).
  
• Try to plot a graph of it's derivative/ slope.
  
• You can either try to do this by eye, or using graph paper and Nascoto's method ("up divided by across" for a little bit of the graph)
  
• See if you can guess what function the derivative of cos(x) is!
  

We'll be waiting

Edited April 4, 2014 by Doozler