Pikachu or Doraemon?

Dresses like Pikachu *Uses Doraemon Song*

dy/dx is not a third grade fraction, it is a “calculus fraction”!

Pikachu + Doraemon music Life is good

dy/dx isn’t a fraction... Or is it? *VSauce Music plays*

bprp : ofc i will not eat the whole pizza tonight me:

I really like this video because it highlights the importance of defining things clearly AND agreeing on those definitions. I've noticed in political discussions that not agreeing on definitions is a frequent problem.

Locally, dy/dx behaves like a fraction; and assuming that dx is infinitesimally small, one can formally write dy=m*dx; and because m is a constant, integrating results in y=m*x+c, which is the linearisation/first order Taylor series expansion of the function at a given point. However, one should be careful when doing so as infinetisimal calculus (treating dy/dx like a fraction) is not without contradictions

People before learning calculus: I have no idea why calculus is so hard, just cancel the d in d/dx to get 1/x People after learning calculus: FML

I saw the poll for this yesterday, and I was super excited for this video. Thanks BPRP!

My friend : dy/dx is a fraction Me: you are not a joker .You are the entire circus.

That fraction explanation reminds me of the half derivative Peyam did... 1+1=2 😂😋

You see, before today, I knew level 1 calculus and, just to understand your videos, I learnt from Product Rule, Quotient Rule, Chain Rule to Harmonic Series to Ratio Test in 3 hours. Must be the most divine part of my life. Writing, as soon as I completed Ratio Test.

Ok but how in the world did you predict 1/e

Oh!!! Thank you for this video!! In fact I have the same difficulties in learning calculus at first when being told that dydx is not a fraction. But it works like a fraction!! Your video is so helpful to me, a graduate student who has done tons of calculus questions, but still, don’t understand the meaning of dydx.

I'm amazed by how well you can explain stuff, keep it up

If someone looks like a Pikachu and talks like a Pikachu, then does it make him a Pikachu? Not always. It could be a Ditto. It could be a clone created by Mewtwo.

I heard that business about dy/dx not being a fraction during my first calculus class ("you can't separate the dx from the dy!"), then saw several years of math, physics, and chemistry professors operate it exactly as if it were a fraction. Just like you did at 9:20. Had once, just once!, one of them had suggested it was a *ratio* and not a fraction, I would have understood it fine. But no one ever did, and I just sat confused the whole time. This reminds me of when you added two integrals together, and I sat flabbergasted over the fact that you just did, while I had years of math teacher tell me "it's improper to do algebra on integrals..." Yet you did and it was perfectly legitimate!

I didn't understand the explanation with graphs, but the part with ratios really got me. It's such a perfect way to show that even though something looks like a fraction, it isn't a fraction.

this needs to become the “send this to your crush without context” video of the math community

10 minutes summarized: Q: Is dy/dx a fraction? A: No, dy/dx is a ratio of two unlike but related things; all fractions are either quotients or ratios, but not all ratios are fractions and all quotients are fractions. Ex 1. 5 mph is a ratio of 5 miles to (per) 1 hour, it cannot be thought of as 5 miles out of 1 hour. It simply wouldn't make sense. Ex 2. Spending $75/$100 is quotient representing the idea that $75 out of $100 was spent, it cannot be thought of as the ratio $75 to $100; in this case this would be saying there are $175 total, which is wrong.

was wondering what was that poll for. now I see that my guess was correct and the poll had a purpose. great video as always!

Thank you so much I had been searching for a video that explained this well for such a long time

In single variable calculus it is similar to a fraction and you can kind of treat it like one (sometimes), but it doesn't work for multivariable calculus.

Oh, wow, I had always thought of dy/dx as a fraction - because I've always manipulated dy/dx algebraically.

Thank you for explaining! I've been doing differential equations and it helps to know that I can use dy/dx like a fraction.

you are really excellent in math and the way you explain solving every problem.

If you want to continue your pizza analogy for the 2/pi as a fraction, you can say your pizza has a diameter of 1 foot, which means you have pi feet of crust. So 2/pi is taking 2 feet of the pi feet of crust.

Dear blackpenredpen in ur sense we can't also say that 8/3 is a fraction becuz we cant take 8 from 3 .dy/dx is a tiny change in y over a tiny change in x so its i think it IS a fraction since its a in physics dx/dt =V and it is a number the tiny change dx is a number infinitly small but it still a number correct me if i am not right

I heard it's not realistic notation but that it helps with the chain rule (dy/du × du/dx = dy/dx) Also you can connect two -1D things (null sets) with a dot.

This is such a nice video, you'll solve people's doubts for years to come! Videos like this which are simple and solve basic questions are very necessary

Love your videos and feeling so helpful.

Very important video to distinguish between ratio and fractions! Thank you ☺️👍🌞 got a little philosophical with the Pikachu but it’s a very good point

thank you. this thing has been boggling my mind for such a long time

Today, 22nd February 2024, after probably eight or so years of watching blackpenredpen, is the first time I learned that the occasional piano intro used in these videos is the introduction to the Doraemon theme.

Excellent explanation!

Good job on the thumbnails and titles lately

blackpenredpen, I absolutely love your videos and your finesse-ful (is that a word?) approaches/explanations/attitudes with math topics - Thank you!!!I do have a comment/question with the results of this video though - Suppose y = x^2. First, I don't think of, say, dy/dx = 2x itself as a fraction, but rather a function (set of ordered pairs). This even applies to the second derivative! However, when we evaluate this function at, say, x = 3/4, we get:dy/dx = 3/2Taking a literal view of an equals sign, can't we say that since 3/2 is a fraction, that dy/dx (when evaluated at 3/4) must be a fraction?Or would that be like saying that since 8/2 is a fraction (I realize it's "improper"), and 8/2 = 4, then 4 must be a fraction as well? Any thoughts?

Thanks so much! This was very helpful!

While it may be a "shortcut" on my part, I'd say a fraction is a number without dimensions, whereas dy/dx has the dimensions of y over x. That's not something I saw much during school in mathematics, but the notion of dimensions is important in physics (in most cases, just checking dimensions consistencies can help notice errors). As such, a fraction would be the same dimension as "1", I believe the term is to say it has no dimension, while the derivative of a position over time has the dimension of a speed. (Angles are, by definition, a ratio of two lengths, so an angle has no dimension, which can be a bit confusing).

that was a brilliant way to explain, thank you

After learning complex integrals in our last lesson, we will now learn fractions.

Physicist: short answer, yes. Long answer, hell yeah

I really like the way he writes by switching markers

By right we should write dy/dx as d(y)/dx but for simplicity mathematicians write it as dy/dx without the brackets.

Is Δy / Δx a fraction? They are not infinitesimals, so that is indeed a fraction. But Δy is not a "part of a whole". But I jest, I jest! Great video, as ever.

Very good explanation. Even I could understand it!

What about 8/2? 8 is not a part of the hole either, since you can't have 8 parts of something into 2. So, thecnically, improper fractions like these are not fractions at all?

So I was just wondering if it's a ratio or not all along, and you just answered my question.

if you set up rigorous foundations to work with infinitesimal quantities then it can be treated as a fraction, but it requires some work to get to that point to begin with (look up nonstandard analysis for more info)

the best Math teacher in KZfaq!!!

Thanks for that. I was going to ask about differential equations where you pretend dy/dx IS a fraction. Off to have my 33/ pizza now!

Can you do a video on how to intuit grad, div, and curl? I'm taking multivariate calc next semester and I have a rough handle on the mechanics, but I have a hard time grasping them conceptually.

In separable differential equations, we treat dy/dx as a fraction. For example, in Newton's Law of Cooling, dT/dt = - k (T -Ta), where dT is Temperature differential, dt is time differential, k is cooling constant, T is body's Temperature and Ta is ambient (surrounding) Temperature. To solve this equation, in the first step we multiply both sides by dt and divide both sides by (T - Ta), as if dT/dt is a fraction.

Thanks Bro. You just answered my lifelong question from 9th grade

Very interresting!

That was beautifully explained

When defining the rational number set, I define the denominator as a natural number, that is {1, 2, 3, ...}. So Q={m/n|m∈Z,n∈N}.

That creepy pikachu smiling at me and waving at me scarred me for life

Oh my god I thought I accidentally clicked on the wrong video when I saw the pikachu question

Answer YES!!! its ftaction for two differentials dy and dx, like f/g for functions.... :)))) and dy(∆x)/dx(∆x)=f'(x) ...differentiall its linear functional..

QQ: What's that music that you have fade-in @1:35? I hear it used a ton over at Ants Canada and I keep trying to find it to no avail. D:

I got an A in calc I and II and totally forgot that the line connecting two points of the function is called a secant line. So I learned more than I hoped just off that one fact LOL

A lot of questions in this video, I like it! So, when we add or multiply fractions they are still fractions, that doesn't change their nature. but adding or multiplying infinite fractions allows us to describe ratios like pi/4 or e^2 etc. should that really change their fraction-ness though?! I mean of course it does but its a good question. I can even add together infinite things and get a finite answer, 1=.9+.09+.009... for an easy example. so whats really going on? I like to think of dy/dx as a fraction when talking about nested functions or functions of more than one variable, of course they dont act exactly like fractions or even ratios for that matter. Its kinda like how most people (i think) approach complex numbers. What can I take from real number algebra and calculus and apply to complex numbers? when do i have to change my definitions to fit new scenarios? I don't even believe in the normal definition for pi really. if pi is the ratio of a circle's circumference to its diameter, AND pi is an irrational number that cannot be written as a/b, someone has been lying to my my whole life! I define pi geometrically as "the area of the unit circle" that's it.

Thank you! Thanks a lot!

you are the best math teacher ever

For youtube 0/0 = 0.5, cause if you look at the likes bar when there are still 0 likes and 0 dislikes the blue part covers 50% of the bar

Todo un genio hermano! Saludos

dy/dx= y/x ... fraction?? *VSauce Music plays*

What's sad about the Indian education system is that it has equipped me to evaluate complex integrals, solve hard differential equations, without actually understanding what dy and dx are.

I disagree. The are many kinds of fractions: common fractions (with integer numbers), but there are also fractions with irrational numbers. The reason dy/dx is not a fraction is that it is the limit of a fraction. Limits of fractions are not fractions. Mathematically a ratio can be expressed as a fraction and in mathematics they are handled as fractions. In wikipedia a ratio is called a form of a fraction. So, a ratio is a fraction in mathematics. Saying a ratio is not a fraction is not correct. en.wikipedia.org/wiki/Fraction_(mathematics)

What if we define a fraction as a division problem? Is dy/dx a division problem? dy ÷ dx?

Thanks sir...

d/dx (y) is a derivative of a dependable variable y with respect to a indepent variable x.It basically it.We express it by the notatiom d/dx (y). But there is another dy/dx which behave as a fraction as we saw countless of time.That dx and dy are called differentials.differentials means infinitesimal change in a variable. They both means pretty much the same in single varibale calculus.but in multibariable calculus there are some differences. One of the key differences is differentials dy/dx means the differential of two function. On the other hand derivative dy/dx means the derivative of y as a function x.

That's a good question and a great answer

blackpenredpen I was wondering: When you compute probabilities and get a ratio as an answer, is that a fraction? Thanks.

Can you explain the mathematical formula for the Anywhere Door?

Can't wait for: is ∫f(x)dx a product?

Is there any operation you can do with a fraction that would break dy/dx? (Not cancelling d's or something like that, have to treat 'dx' as one symbol) Is this correct: y(dx) = dy? I explain dy/dx by saying it is like having a rectangle with height and width but now scale the rectangle down arbitrarily small and you can no longer say how big the rectangle is but the ratio between height and width remains constant. Is this a good explanation?

And here I was thinking that a fraction was an equivalence class defined on a ring R with (a, b, c, d) in R (a, b) ~ (c, d) ad = bc and b, d are not zero divisors. :(

I noticed BPRD never truly justified why dy/dx is a ratio, other than just saying "it doesn't make any sense" and proclaiming that dy/dx is similar to a ratio though analogy - that is not a proof that it isn't a fraction. All fractions can be expressed as ratios, and saying dy/dx can be expressed as a ratio or that it is similar to one doesn't rule out the possibility that dy/dx is a fraction. Change my mind

BlackpikachuandRedpikachu???

i love this video. I was so confused why do we do "du" in U-substitution, thanks man.

Sir u are great.

what an unexpected question !!

If, for vectors, you can say that 2v/v=2, then it is the same thing with dy/dx, and then it is a fraction

If i have a complex number = 1 + 1i , then i devide it by 2 (a real number) then it become = (1+1i)/2r = 1/2 + 1i/2 I believie the (1/2) part is a fraction because both lay on the real axis My question is the (1i/2) part, or we simply wrote it (i/2) in math, is it a fraction? If i define it as [(1/2)* i ] i believe the (1/2) is a fraction then it multiply by i If i define it as (1i/2) would it become a fraction as well? Because it's not lay down on the same axis which mean imaginary numb. is not part of the real numb. , what do you think?

Your explanation is amazing notwithstanding your English (I'm trying to figure it out btw)😅😅

The prediction was amazing....

Love your shirt. Where can I get one like it?

Isn't implicit differentiation (dx/dx = 1) just the inverse property of multiplication as applied to differentials?

Further confusing the matter is that we tend to stick dy/dx in front of some expression as if it's part of a product. Then you take linear algebra and vector calculus and we are placing all sorts of symbols and operators in front of other symbols, the way we learned to write products, including but not limited to: matrices multiplied by matrices, matrices multiplied by vectors, and functions composed with other functions (which, we are told, are equivalent to the transformations given by a matrix). And then our good old friend dy/dx is there too, along with its bizarre cousin ∂f/∂x, and of course in all of this mess there are several types of product, some with their own symbol, and some taking no symbol at all...😝

"Is dy/dx a fraction?" Me: Is this a trick question?

Im a dirty physicist. obviously I use it like a fracti0on, makes life a lot easier. Thought the mathematicians at our university hate us, for using such dirty techniques.

Define the differential df(x, Δx) = f'(x)*Δx, where f' means the limit definition of the derivative. Then dx = dx(x, Δx) = Δx, so dy/dx = (f'(x)*Δx)/Δx = f'(x).

couldn't a fraction be defined as a ratio of one thing to another? which in this case it'd be the change in y to the change in x?

What if discrete geometry, where dy,dx belong both to Z? If in CAD drawing grid is set to 1.0? Y and X easily can be y and x coordinates.. but what if da/db? bu

I've got a question: If m = tan(a) where a is the elevation angle of the line, then could be dy/dx = tan(a), a being the elevation angle of the tangent line? Is this in some case useful?

Great video! But isn't 8/3 also not a part of a whole?

I want to know why the variable separeble method works