Visual derivative of x squared

A visual of the derivative of f(x)=x squared. We show how to think about the derivative of a function visually.
#manim #calculus #derivatives #derivative #tangentline #slope #parabola #mathvideo #mathshorts #math #visualmath #graph #parabola #linearapproximation

[@kirahen0437]

I'm 14 and I haven't yet done calculus, but I hope this explanation will come in handy when I finally get to it

[@ysnsmth]

to be honest its more on shortcuts and equations rather than actually looking at graphs but thats my experience (speaking as someone learning these rn)

[@horu1688]

if you are interested into calculus, it’s easy to get into it and learn

[@kirahen0437]

@@horu1688 thanks, I am currently studying trigonometry on my own, because Italian school does calculus and trigonometry only in 4th and 5th year of highschool. I've heard that you need to know trigonometry before going on to calculus. But it's very hard for me to memorize all the formulas. I hope I can learn calculus soon, because it's one of the only things I'm actually passionate and exited to learn about. :)

[@bigchungus1618]

start now its fun

[@littlegrass320]

​@@kirahen0437 for memorizing formulas spend time understanding how they work and practising, u wont need to memorise then it will just come naturally.

[@chilldo5982]

Another interesting way to see it, is that if you look at the intervals between the perfect squares (1, 4, 9, 16, 25...), you can see that the intervals are 3, 5, 7, 9... The odd numbers! And the odd numbers are represented as 2x+1. But we will leave the 1, because of derivative rules which I cannot explain because I don't know it myself too much.

[@lukandrate9866]

It's because (x+1)^2 = x^2+2x+1

[@user-pr6ed3ri2k]

isn't 2x+1 the discrete derivative of x² on the integers?

[@squeezy8414]

@@user-pr6ed3ri2k Yeah it's known as the forward difference, very cool analogy but not quite the same thing as the continuous case in calculus

[@raygolu]

You can remove 2 for general solution 😅

[@Lukas-vs9df]

Derivitave of 1 = 0

[@lastoneleftmedia7274]

If teachers taught like this instead of teaching us "tricks" to solve the equations, it would not have taken me 30 years to finally understand derivatives.

[@sumithadavid5841]

I was searching this explanation for nearly 2years and now I understood Thank you so much, 😊😊😊😊😊😊😊😊😊😊😊😊

[@hiimgamerspruzzino5804]

Sometimes all we need is a very simple explanation. I've been studying derivatives at school since the last few weeks and couldn't really understand them deeply. Thank you so much!

[@izzabelladogalini]

Excellent graphic representation of derivatives

[@karthikteal1016]

I learnt that 13 years ago I understood the meaning today

[@Aditya_196]

Basically most people forget the essence of derivatives.... Derivative means rate of change over a small interval ( very small : dx ) So imagine a square with sides x its area is x² now if u increase its dimensions from x to x + dx its area is x² + 2x*dx + dx² This means its area increased by 2x*dx + dx² .... Now this change with respect to that small interval dx means the expression 2x*dx + dx² / dx = 2x + dx Since dx is very small ( 0.00000....1 ) we ignore it So d ( x² ) / dx = 2x

[@Aditya_196]

Note : the derivative is defined as the change over a very small value It is given by the expression d f(x) / dx = f(x + dx) - f(x) / dx Here dx means a very small defined value so dx ≠ 0 not →0 but something defined and means to be a small value ( a very small change value inputs available to f(x) ) so u can't say its 0 if somewhere it is written dx it varies from function to function... In simpler terms the dx of x² and e^x and lnx are all different

[@tonitalas1757]

Nice visual😊

[@MathVisualProofs]

thanks!

[@_PsyChoPaThiCX.___.]

can you do the same for integration plzzzzzz

[@MathVisualProofs]

I’ll see what I can do. This one didn’t end up as I want (aspect ratio) but maybe I can figure it out :)

[@Khalid.115]

The integral of a function f(x) is just the area between that function and the x-axis. This has always helped me solve integration problems. I'd love to see a video about that too, assuming there isn't one already.

[@ploppen3872]

@@MathVisualProofs man I did literally not realise something was wrong. Great video

[@kbin7042]

Think about integrals like this (i recommend trying to draw as you read) Let's say you have the velocity of a particle in relation to time, the function V(t), and you wanna know the area of the surface created between the graph of V(t) and the X axis in a span of time Let's also say that you know a function that can give you that area for any span of time, this function it's called an integral. obviously this function will be dependent of time so we'll call it P(t) (more about it later) If we go infinitesimal, we can think about the surface we want as if it was made of infinite bars, bars that are the segments created between each instant t and it's corresponding V(t), all of them glued together. In an infinitesimal scale , if we change the span of time by just one instant( one point in the X axis), we'll definitely change the surface created, but we can see that change as if we were adding a bar or taking a bar out of the surface. Now , this bar has basically no width, but it has height, and the height it's obviously the same as its corresponding V(t) , so we can say the bars we add or take out are actually one dimensional segments of height V(t) that make a surface just because we have an infinite number of them So here the final conclusion, since the rate of infinitesimal change of the function P(t) (the area of the surface) in relation to t is determined by the bars we add or take out, in other words the segments of length V(t) ; we can say that we've efectively defined V(t) as the derivative of P(t) in relation to t, by the very definition of what a derivative is And that's how we conclude that the first derivative of the position is the velocity, a very well know relation, and we also discovered that an integral is not only an area, but also an antiderivative of the function that we evaluate

[@AkamiChannel]

@@MathVisualProofsthere's something about archimedes realizing the integral of f(x) = x, or something like that. Not hard of course. Maybe a good video topic tho. U can folloe his same method

[@ВладимирКосицын-с6о]

The most underrated math channel, what i've seen. Now i can explane how to woks derivates for myself.

[@itsnotshakespeareitsdennis6877]

Omg! How beautiful is this presentation!!!!

[@JBroadway]

bro unlocked calculus in my brain wtf

[@jackgately7992]

makes so much more sense thank you!

[@40fpskiller]

Thank you so much. i always need such type of explanation.

[@tim40gabby25]

Nice job. Visuals are so helpful. Curious old UK duffer here :)

[@christianmosquera9044]

Excellent video

[@MathVisualProofs]

Thank you very much!

[@ssaamil]

In love with the visuals, what program is this? Amazing explaination too :>

[@MathVisualProofs]

I use manim for these.

[@sahandfars7009]

Very useful thx

[@NECTOMIC]

Finally I have known what is a derivative

[@GamerGod_99]

Cool

[@blueman6557]

So good

[@aram5642]

why does everybody explain derivatives on an x² example. it is so deceiving

[@MathVisualProofs]

I have some others in the queue. This is the first and easiest nonlinear example.

[@RazzleberryHaze]

Because it's a good basis for introducing someone to the concept of differential quotients.

[@brainloading5543]

Because it's one of, if not the, simplest slope(s) that is not a line.

[@asparkdeity8717]

Good intuition for people who’ve never come across calculus yet (especially the kids)

[@SebastianRodriguez-rk4vx]

I’m five minutes into learning derivatives and this has taught me more than any teacher

[@srikrishnaghosh7660]

Man! you are going to kill the imagination of young calculus lerners😅 jokes apart, we had to visualise this same idea in our mind only, and now kids will learn it easily. Beginners will find this very helpful and will use their time to think about something more abstract ❤ i am going to use it as a teaching material 🎉🎉🎉🎉🎉thanks for sharing.❤

[@mahamamjad8150]

I really want to know how is that -7.82 and 1 came . And you said is we compute slope of the function along the curve but that would be an equation...... How do we get those points that you plot in the below graph?????

[@MathVisualProofs]

Check this: kzbin.infoeE4IGCAzmqA?feature=share

[@mahamamjad8150]

@@MathVisualProofs thanks alot ✨🦋🦋🦋🦋

[@mahamamjad8150]

And the visualization of mathematics is amazing i love exploring details in geometry. So your videos made sense to me. Thank you ✨🦋

[@NumbToons]

Dude, I have this video/visualisation recorded in my mind (i mean before watching your video)

[@Dan-1031]

I always knew x^2 is 2x but I never knew the reason as to why, I was just given a formula with no context

[@hartobks3466]

Thanks,

[@OC74.]

That’s soo cool

[@riki_XD]

Yup my mathematics teacher sucks This makes easy to understand Thank you for making such videos

[@AntonioBarsanio]

What software do you use to play with those graphs? Pretty Neat and also easier to understand the concepts.

[@mohammadsadeghabedian1974]

fascinating

[@tristancole8158]

Do the infinite series of ⅔+4/9+8/27… next!

[@MathVisualProofs]

I think That sum is hidden in this video (kzbin.info/www/bejne/raCuh5trZrCDopY ) in terms of side lengths if you assume each side is length 1.

[@tristancole8158]

@@MathVisualProofs woah, that’s your best video I’ve seen by far

[@EvilDudeLOL]

I am 14 years old and I am about to finish calc 2 by myself (don't judge me I am just a curious child). Going back to the definition is always refreshing. Great job sir.

[@therussianbullfrog3975]

You will be judged and heavily so...

[@pritamyadav17]

Nice

[@Bmb8007]

you could use the power rule to get the derivative without the graph if given the equation, but it gets tricky when there are more complex exponents like fractional or negative exponents

[@ethanhunt6120]

This is for people that are very early Calc. Or don’t know Calc at all

[@walkdead94]

I wish my teacher had taught that at school

[@sumithadavid5841]

But the linear equation of the 2nd graph can also be “ f’(x)=x or f’(x)=nx where n belongs to natural no.s” Does that mean x,2x,3x…bare the derivatives of x^2???? PLESE CLEAR THIS!!!!!!!

[@moocowpong1]

I wonder if there’s any interesting relationships you can show visually involving Legendre transforms.

[@_rd_5043]

Is this a re-upload because there was y=x instead of 2x before?

[@MathVisualProofs]

It was y=2x but I can’t get aspect ratio right. So I added labels :)

[@_rd_5043]

@Mathematical Visual Proofs it's better with the labels honestly

[@MathVisualProofs]

@@_rd_5043 thanks. Still doesn’t look as steep as I’d like.

[@_rd_5043]

@Mathematical Visual Proofs it may still be 45° but at least the labels make it clear. And some functions are so steep you have to squeeze them anyway, so it doesn't really matter.

[@nascimentomoreira9668]

Derivative of ○r² ~ 2○r ... A ~ C 4/3○r³ ~ 4○r² ... V ~ S ○x² .f(2r) = 4○r² Area Circunference Volume Surface

[@nascimentomoreira9668]

Can you teach please how to calculate the angle that way you show in tha class

[@qiangli1578]

which kind of curve can be derivative? Are there some characteristics on it?😊

[@SistemDışı-z3l]

And drivative of x³ function is parabolic.

[@날개달린_양]

wow ty

[@anatolykarpov2956]

That's great , I hope you make another one showing the role of limits here were the tangent line is obtained after a sequence

[@gonnguyen2341]

Hi teacher, How can you draw these visual graph. Do you use special software?

[@MathVisualProofs]

Yes. I use manim. This is the python library from 3blue1brown

[@gonnguyen2341]

@@MathVisualProofs THanks sir.

[@noname-kh7sg]

Hey you missing something, add those classical songs

[@MathVisualProofs]

Some people say not to have music. So I’m trying both. Thanks

[@lumnisxate_192]

Can i ask something??? Where can we use the instantaneous speed?? Like d/dx x² --> 2x f'(2) = 2(2) = 4 Meaning the instantaneous speed at point 2 is 4, what does that mean?? The next point will increase by 4???

[@olixx1213]

If you zoom in at x=2 , the curve will turn into a straight line This straight line has a slope of 4

[@lumnisxate_192]

@@olixx1213 Ohhhhhhhhhhhhh!!! I see!!

[@AnglandAlamehnaSwedish]

Ahh i miss me TI 89 paid alot of money for it 26 yrs ago like $90

[@good4455]

I don't understand and no one tells me WHY the tangent line lenght changes? Why does it matter if it's shorter or longer it's the same idea

[@MathVisualProofs]

Not length. Measuring the slope. I just had to draw a segment of the tangent line.

[@ngocphuonghoa7622]

What software can I use to make this video?

[@mahamamjad8150]

Which kind of coordinate axis you choose I'm struggling so hard to understand how did you marked y=7.82 ? Which x values are you choosing?

[@MathVisualProofs]

The x-axis ranges from -4 to 4 and the y-axis ranges from 0 to 16. I didn't start quite at x=-4 though with the dot.

[@νίοιετ]

@@MathVisualProofs so basically we have to ignore the numbers ?

[@MathVisualProofs]

@@νίοιετ the numbers are The slopes of the tangent line. Can’t ignore them. That is what the derivative measures.

[@νίοιετ]

@@MathVisualProofs no I'm asking about are they accurate ? It u just used it for example ? Does those value really correspond to x and y axis if I go to do calculation ?

[@MathVisualProofs]

@@νίοιετ these are accurate. The short doesn’t show how to compute them (longer vid needed) but if you run through x values, the tangent slope will always be twice the x-value.

[@aaryalamichhane810]

how is this video made?

[@MathVisualProofs]

Manim

[@weum.]

Where were you when I was in high school?

[@StandDont]

me in 10th grade: 💀

[@pastaplatoon6184]

And just like that derivatives make 1000x more sense.

[@theoverseer393]

Acceleration is a derivative of Velocity

[@abdullaalsulaiti1803]

Velocity is a derivative ot displacment as function of time

[@ericwilliams1832]

You are a derivative of acceleration!

[@molybd3num823]

​@@ericwilliams1832 nah why you saying that :v

[@vinayak6237]

pls like if anyone understood

[@Saihnaa67]

Well iam at only 6th grade and I know trigonometry so well and calculus and iam in advanced maths grade but I didn't learn trigonometry at my class but I watch math and learned my math teacher really proud of me

[@tunistick8044]

ok?

[@DeathrashWhiplash]

Stay with it young one do what is true to your nature

[@Rangadus]

to be fair, if you just find the derivative using the power rule or chain rule, you can plot these functions very easily, d/dx x^2 is obviously 2x, so you just need to plot the function f(x)=2x.

[@jamieelaalbarsha9302]

ماهو البرنامج المستخدم

[@MathVisualProofs]

manim (on top of python)

[@jamieelaalbarsha9302]

@@MathVisualProofs ❤️❤️❤️❤️❤️

[@홍삼깨보리]

JYP 춤 YG 춤 SM 춤 HYBE 춤

[@IreneLenz]

guess what i know understand what does derivative mean

[@Japeshvundavalli]

what do u took for x axis u got slope 2 u took x axis as x^2

[@Saikat_Musib]

I understand nothing!

[@Yeastboi]

you lost me at consider

[@opalkarli]

this makes no sense

[@tunistick8044]

it makes so much sense

[@DeathrashWhiplash]

It is beautiful once you understand it friend

[@eliseenkoivan]

Blyat’, this is brilliant! Now I understand why is this a useful thing.

[@much_tunsia]

Looking for same video witj second dev. 🥹🥹