Check out how to differentiate the quadratic formula: kzbin.info/www/bejne/gHbGdmCtebeAec0si=Uq1p1KYNj-Kpu5g2
@KluhlHloo11 ай бұрын
Integrating the quadratic equation = god mode! You sir are a badass!
@abishworpandit520011 ай бұрын
Well everything has physical meaning so what does it physically mean?
@KluhlHloo11 ай бұрын
@@abishworpandit5200 🤔
@1Konu1Zoru8 ай бұрын
dont rely on AI and dont ask AI to any math question.. it is very dumb in that area and still thinks that it knows the answer.. but in near future it will eventaully learn to do math and most likely beats any human being on earth :) wolfram is around for a long time and it is best tool to check results or solve something for the timebeing.
@BradleyG0111 ай бұрын
If there’s one thing that ChatGPT tells us it’s that AI won’t be replacing math professors anytime soon.
@aashsyed127711 ай бұрын
Well that may not be true in the future....
@resurrectedpa11 ай бұрын
@@aashsyed1277won't be soon 😅
@acuriousmind621711 ай бұрын
there's already multiple AI's online that can integrate any integratable function and give you the full meticulous steps idk if you count that as replacing math professors.
@littlegrass32011 ай бұрын
@@acuriousmind6217 a math professor does a lot more than integrating functions, they deal with a lot more abstract concepts that don't have a set way to do them. Until those AI's can understand abstract concepts, math professors won't be going anywhere
@teelo1200011 ай бұрын
ChatGPT write me a witty youtube comment reply.
@badamson11 ай бұрын
If you change the last integral (1/(u^2-b^2)) with diff of two squares you can split it up with partial fractions and use natural log… although it turns out the end result is actually an identity for inverse hyperbolic tan
@Ninja2070411 ай бұрын
They aren’t entirely the same because of the domain difference. tanh^-1 (x) is only valid for -1
@bobbyheffley495511 ай бұрын
@Ninja20704 if x>1 or x
@liamernst962611 ай бұрын
w pfp
@mrfarooqkhan845411 ай бұрын
@@Ninja20704bro can you explain why sometimes we get two different answers for the same integrant? How to decide which one is correct?
@Ninja2070411 ай бұрын
@@mrfarooqkhan8454 Usually it is because the two different answers are actually just off by a constant, so they are essentially the same as far as the indefinite integral is concerned. One example is secx*tanx. In this case, the different answers we are getting is more of domain, but the functions have the same value for all x in the common domain. The integral of 1/(1-x^2) has 3 seemingly different answers: tanh^-1 (x), coth^-1 (x) and ln|(1-x)/(1+x)| tanh^-1 (x) = 1/2*ln[(1-x)/(1+x)], which is only good for -1
@CorrectHorseBatteryStaple47211 ай бұрын
I love the grin at 6:03 with the inverse hyperbolic tangent. Math is fun and your subtle humour like this makes it fun.
@mrizwan482811 ай бұрын
😅
@skyethebi11 ай бұрын
5:28 You can also use partial fractions for this integration u^2 - b^2 = (u - b)(u + b) 1/(u - b)(u + b) = 1/2b(u - b) - 1/2b(u + b) And that way you don’t end up with inverse hyperbolic trig
@xninja236911 ай бұрын
I was gonna write sameee
@perfectman307711 ай бұрын
Nerd
@xninja236911 ай бұрын
@@perfectman3077 stupid Sigma 10 year old kid , your place is not here go somewhere else dude 🤡 If you want to learn something than learn, if don't than get out ..
@joyboy69lffy11 ай бұрын
so u will get 1=A(u-b) +B(u+b) so you cant find the value of A and B since you dont know neither the value of u nor b
@skyethebi11 ай бұрын
@@joyboy69lffy My bad. You can still use partial fractions since b is constant with respect to the integration but you need to divide by b. I updated my answer so it should be right now.
@acuriousmind621711 ай бұрын
A much more interesting approach is differentiating each quadratic formula with respect to each variable (a, b, c) and observing how each variable contributes to the fluctuation and the identification of the sweet spot for every variable that results in a more diverse set of solutions.
@dekippiesip11 ай бұрын
Sure, a lot of cool things could be done with this. Take the solution set as a function with 3 variables, include complex numbers if you like, and look for all sorts of patterns. I'm thinking of stuff like maximizing the distance between 2 solutions for all vectors (a,b,c) of fixed length and shit like that. It's a heavily unexplored topic!
@ethancasillas732511 ай бұрын
There is absolutely no reason for me to have watched this, I'm not in -or ever going to take- any calculus class where I have to integrate the quadratic formula. Yet here I am, loving every second of this video :)
@blackpenredpen11 ай бұрын
Thank you!
@ahmedalhomaide441611 ай бұрын
Bro, I just graduated from high school but I can't ignore your video titles... Great as USUAL!
@blackpenredpen11 ай бұрын
Thanks!!
@MathNerd172911 ай бұрын
The last part reminded me of when I asked 2 different AI models (not ChatGPT) about whether they can prove Euler's Sum of Powers Conjecture . . . both of them had factual inaccuracies in their responses and one of them even claimed that it's currently unknown
@acuriousmind621711 ай бұрын
its the same as asking google a long question... at the end of the day its trained on pattern recognition you have to de-personify the ai so to speak when approaching it with complicated questions
@decaydjk892211 ай бұрын
All they are is token prediction systems, just a more advanced version of the predictive text in your phone. They can't "do" math or anything so no, that's not surprising
@abuabdullaahiwaaaishatah823511 ай бұрын
Lol
@lolerie11 ай бұрын
Because only GPT 4 is AGI. Above human level.
@sunnohh11 ай бұрын
Yeah ai is dogshit at medium hard or slightly complex
@abyzdoof882111 ай бұрын
i love how we can just see all of those dry erase markers on standby in the bottom right. video’s fantastic as always, keep up the great work!
@jamesdcarroll111 ай бұрын
Its been 30 years since I took this in college and I can still fall asleep halfway through.
@Alkis0511 ай бұрын
Who said you would forget how to do the things you learned to do in your freshman calculus course?
@dayzimlich11 ай бұрын
Great stuff, bprp! I love your videos. You teach but you do not talk down to the audience and you bring a catchy sense of humor and fun. Here's to your continued success!
@blackpenredpen11 ай бұрын
Thank you very much for your nice comment!
@blackpenredpen11 ай бұрын
Proving the quadratic formula while blindfolded -> kzbin.info/www/bejne/fYjVoph8eqZ2nrssi=AubYmfac-FODRJVj
@aaronyoung587111 ай бұрын
Damn
@thatomofolo45211 ай бұрын
Yo 👋
@That_One_Guy...11 ай бұрын
Can you differentiate and integrate the cubic formula ? Or even the quartic formula ?
@youssefbenmMorocco11 ай бұрын
But why?
@mathwizardmathwizardmathwizard11 ай бұрын
@@youssefbenmMorocco Because it's fun.
@tuseroni608511 ай бұрын
that trick of adding in a -b and a +b blew my mind. it makes perfect sense in retrospect, -b+b=0 so if you have something that is also something + 0 and thus you can put anything in so long as it cancels out to 0...
@blackpenredpen11 ай бұрын
Yes 😃
@gamingbeast575511 ай бұрын
Math Professors are the real JOD people on Earth....
@unonymous-2711 ай бұрын
Its a fact that ChatGPT has cracked the toughest exams but does dumbest mistakes in new type of questions....
@elquesohombre993111 ай бұрын
Those exams aren’t usually logical ones, that and it will have many resources that allow it to “cheat” on those exams by pulling data from sites that have very identical questions/answers
@jok200011 ай бұрын
Just ask it to write a python program to symbolically integrate.
@SlipperyTeeth11 ай бұрын
Me holding a stack of answers to every exam ever published online: It's a fact that I can answer the toughest exam questions, but am unable to apply any reasoning skills to "new types of questions".
@brookek311611 ай бұрын
@@elquesohombre9931Yep, I’d do pretty good on a lot of exams too if I was able to copy and paste answers from the internet
@Craznar11 ай бұрын
Given it is only trained on language, that it even attempts to work out the answer is impressive.
@zelda1234611 ай бұрын
"I integrated the quadratic formula" has the same energy as "I sawed this boat in half!". And since you differentiated this already, "I sawed ANOTHER boat in half!"
@pulkitgupta536711 ай бұрын
Its giving the same energy as that flex tape ad where he cuts a boat in half and then tapes it back together: kzbin.info/www/bejne/Zqndf2l8gpqrlac
@DatBoi_TheGudBIAS11 ай бұрын
ive figured out a long time (by curiosity) dat chatgpt absolutely sucks with math problems. it can fail in something as simple as a quadratic equation, let alone a full integral
@winghei1011 ай бұрын
Teacher: Maybe there is a mistake. Please verify the answer by differentiate the answer.
@MayurAvad11 ай бұрын
Really enjoyed this one !
@blackpenredpen11 ай бұрын
Glad to hear. Thanks!!
@panhandlejake620011 ай бұрын
So definitely an academically interesting exercise. Is there any insight about quadratics and their solutions that can be learned from this result?
@dhruvaraju587011 ай бұрын
No!
@thbb111 ай бұрын
Beautiful, but can we find a geometric interpretation of this integral? Or at least use it to gain some insights on the regions where the determinant is positive or negative as a function of the first coefficient of the trinome?
@gileadedetogni905411 ай бұрын
Maybe f(a)=all the formula
@Jordan-zk2wd11 ай бұрын
Someone correct me if I'm wrong, but utilizing the intuition of the mean value theorem I think this means: if you vary a from some value a0 to a1, then the mean value of x which solves ax^2+bx+c=0 will be [f(a1)-f(a0)]/[a1-a0], where f(a)=-(b/2)ln|a|+sqrt(b^2-4ac)-(1/b)tanh^-1(sqrt((b^2-4ac)/b)* . I tried running it on desmos, I'm tired so I might have did this wrong, but here's an example of what I mean: from a=1 to a=2, the mean value of one of the roots of ax^2+3x+5=0 is -2.66324ish. I did have to throw in a negative sign at the end to make it all work, which means I'm made some small error along the way, but visual check of a few examples seem to confirm that it does work. * Important caveat, this will be the average value of one of the roots as you very a, the other one would have a slightly different definition of f(x) for the choice of the negative root of b^2-4ac, and thus give you a different average value corresponding to that root. @blackpenredpen ?
@ivanrodionov972411 ай бұрын
Is there a physical interpretation to integrating the quadratic formula with respect to the polynomial? Does it tell us anything special?
@spearmintlatios904711 ай бұрын
If we were to create a graph with an a axis and an x axis, where x is vertical (where y usually is) and a is horizontal (where x usually is), this would tell us the area under the curve of the graph. x = (-b+sqrt(b^2-4ac))/2a in terms of b and c which are constants. That’s really it, it’s not too special. If you wanted to see what this graph would look like you could go on Desmos and substitute x for y and a for x, and then create sliders for B and C.
@ivanrodionov972411 ай бұрын
@@spearmintlatios9047 Well that it can be viewed as an an area under a curve is one interpretation, however this is not what I am asking. He is doing something much more fun here, see, the quadratic formula is a function in polynomial of 2 degree space which has 3 degrees of freedom, a,b,c. It therefore maps a unique polynomial represented by the point (a,b,c) in the space and maps it onto two values which are it's roots. Now, when he integrates the quadratic formula, with respect to a, this is the question on how it changes the mapping and what does it tell us now. I am 99% sure there is quite a bit more going on here.
@spearmintlatios904711 ай бұрын
@@ivanrodionov9724I kinda get what you’re saying but I’m rusty on calc 3 stuff tbh. If you want, you can represent the quadratic equation as a graph mapping the function of a, b, and c to a value of x. or f(a, b, c)=x. this would be a 4 dimensional graph which is somewhat beyond our imagination, but everything should still apply. Since we only integrated with respect to A here, B and C are still able to be changed within the integral. pretend that B is equal to 1 for now. Then we can have a 3D graph of f(a, c) = x. In this case, we have a the a and c axis representing our domain, and an x axis representing our range. So for any point on the (a, c) plane, the output of this function is some “height” in the x axis creating the point (a, c, x). If this is the case, and we assume B is constant, then this integral represents the area summation of the height from the x output to the flat ac plane from a given start and end point a0 and a1, So it would still just be a 2D area found from an infinitesimal slice of a 3D graph. It is cool that we could move this slice by changing our bounds of a0, a1, and C. You could imagine changing these bounds as moving and stretching a tall piece of paper through the 3D space of the function, and calculating the area of that paper if you slice off all negative values and everything above the x points. To calculate the volume of the shape this graph makes, we would have to integrate by another bounds as well, a double integral. But the same logic works as well when we include B again. The integral would simply be a slice in a 4D shape that has X as the output. Unfortunately, we can’t just set X to 0 and have a graphical interpretation of 0 = integral (g(a, b, c))da. I don’t think there actually any representation there? If we wanted to find an integral that works here we would probably have to move some terms around before hand and treat it as a differential equation. Correct me if I’m wrong
@spearmintlatios904711 ай бұрын
@@ivanrodionov9724 after reading what you’re saying again I see you’re talking about something entirely different. Is there any tool that could create a mapping similar to what you’re talking about? I think what I was discussing might relate when you consider he only integrated using a plus and not a minus
@minseok872611 ай бұрын
@@spearmintlatios9047any applications in Engineering could you think of? Aerodynamics maybe
@gachanimestudios834811 ай бұрын
Legit the first question i asked from that title was "with respect to what?"
@jok200011 ай бұрын
If you want ChatGPT to do advanced stuff you tell it to write a python program that does what you want like plot a function, do a symbolic integration or even get it to create a rotating globe in OpenGL. I've done all of these things with it and more... a subway station mapper using google maps API and TKinter etc. Some, like the maps API require a bit if tweaking but not the integrator.
@granthqweqw524411 ай бұрын
Would love to see triple integration with respect to all variables there!
@mynamejeff6930211 ай бұрын
Last part of integration with u^2-b^2 can be done with writing it as( u+b)(u-b) and then matching the numerator to get 1/2b^2 ln(u-b)/(u+b)
@kranhat11 ай бұрын
0:10 beep sound
@vaccino33599 ай бұрын
Applied maths: Useful things that would probably come handy in life Pure maths:
@TranquilSeaOfMath11 ай бұрын
Love the video. Easy to follow; well explained. I also like how you show the technology failure at the end.
@adrienanderson743911 ай бұрын
Another one that's fun is the limit as a goes to 0, It gives an interesting result that makes sense when you think about it. If you really want a challenge you could do the limit of the cubic formula as the x^3 coefficient approaches 0 to see if you can get the quadratic formula
@euler111 ай бұрын
I did that like 2 month ago and thought it was very nice how the limit works out at the end. I don't know what you mean with the coefficient of x^3, what equation will you apply the limit to?
@adrienanderson743911 ай бұрын
@@euler1 there is a cubic formula but it is kind if complicated
@adrienanderson743911 ай бұрын
@@euler1 The solutions to ax^3+bx^2+cx+d=0 is given by the following: let u=b^2-3ac, let v=2b^3-9abc+27a^2d, let w=((v+-(v^2-4u^3)^(1/2))/2)^(1/3) where we can use any root for each cube or square root. Then x=(-1/(3a))(b+w+u/w) gives the roots
@euler111 ай бұрын
@@adrienanderson7439 It's the first I see this thanks for the explanation, I am curious to try it out but I think I will just look up a proof for it as I don't have that much time xd
@adrienanderson743911 ай бұрын
@@euler1 Yeah honestly I haven't done it myself and I tried to today but it really gets into the weeds
@SomeoneCommenting10 ай бұрын
It would be interesting to compare the integrals of da, db, and dc for the same curve parameters to see what happens.
@holyshit92211 ай бұрын
6:09 If u is in interval (-1,1) but if u is outside this interval you have b/u as argument of inverse hyperbolic tangent
@gamingbeast575511 ай бұрын
Nice Explanation Sir...
@NonTwinBrothers11 ай бұрын
Im glad I woke up at 3am to enjoy this 😂
@disgracedmilo11 ай бұрын
the sequel we needed
@francaishaitam67084 ай бұрын
for the inverse hyperbolic tan part,you've better to go with a partial fraction decoposition ,since we don't now the domain of integration , it's ok to go with the inverse hyperbolic tan only if a is from (-1,1)
@arsalmathacademy11 ай бұрын
I am always fan of this great person, brilliant brain
@arsalmathacademy11 ай бұрын
@@English_shahriar1 Thanks
@fredrickfred46217 ай бұрын
You can also use trig sub to avoid hyperbolic tan
@promethius782011 ай бұрын
This integral is basically the unbounded interval is basically the terms for the "missle knows where it is" problem, isnt it?
@A-0II0Io11 ай бұрын
😂😂😂 the last part of your amazing video was amazing. Wow...
@honestabe_920711 ай бұрын
The solution for either case (+/-) excluding hyperbolic functions when integrating with respect to "a": -b*ln|√((b^2)-4*a*c) ± b| ± √((b^2)-4*a*c) + C
@OleJoe11 ай бұрын
This is wild, man!
@adarsh599711 ай бұрын
5:39 but the integrals of the form dx/(x²-a²)= 1/2a*ln(x-a)/(x+a) where ln is natural log
@jannegrey5937 ай бұрын
Have you ever tried differentiating/integrating quadratic formula? Yes, basically as soon as I learned how to do basic derivatives from limits and noticing that circle/sphere calculations look like they have been integrated/differentiated using "power rule" (I think that's what it's called). 2*pi*r, pi*r^2 and 4/3 pi r^3 looked too much like coincidence. So I looked at quadratic formula (I also wondered back then what would happen if I did look for zeros when "delta" is "negative" - of course without imaginary numbers it is "harder" to do. Still doable in some cases and if you don't mind going a bit outside real numbers for rules). Sadly my knowledge was insufficient - had I tried it when I was more proficient it would likely help.
@merixan932211 ай бұрын
I literally just watched the differentiating video of the quadratic formula yesterday
@Tasz_11 ай бұрын
Came for blackpenredpen got blackpenbluepenredpen what a day to be alive
@sless692811 ай бұрын
I'm just amazed you found 3 working whiteboard markers.
@Alkis0511 ай бұрын
Wolframalpha still has a lock on the freshmen calculus student market.
@General12th11 ай бұрын
Hi Dr. Pen! Very cool!
@blackpenredpen11 ай бұрын
thanks!
@mqb3gofjzkko7nzx3811 ай бұрын
I just noticed the hundreds of black and red markers in the bottom right corner.
@blackpenredpen11 ай бұрын
😆
@dr0g_Oakblood11 ай бұрын
Last segment is a perfect example of ChatGPT just making stuff up lmao
@said_qurbanov11 ай бұрын
Bro will integrate my life
@Noobthepro011 ай бұрын
Instead of using inverse hyperbolic function as the integral of 1\(u^2-b^2), you may also use ln{(u-b)/(u+b)}/2b. That's how GPT might have also got the answer in terms of natural log
@inverted25337 ай бұрын
you know what, im startin to like this guy
@arghamaji823411 ай бұрын
Yes I have integrated the sridharacharya formula
@Krish-su4oh11 ай бұрын
Last part was very meaningful 😂😂❤
@muralidharrangaswamy964311 ай бұрын
I think you can also expand the denominator of the last integral into partial fractions and then express the result as a difference of two logarithms
@pro_learner.sohamkakkar11 ай бұрын
Wouldn't it be better to do trigonometric substitution in the 2nd integral instead of substituting the root as an algebraic variable?
@berenedain842711 ай бұрын
"He's too dangerous to be kept alive!"
@DeGuerre11 ай бұрын
Not enough people seem to know that the quadratic formula for p(x) = ax^2 + bx + c is: (p'(x))^2 = Δ where Δ = b^2 - 4ac is the discriminant.
@MathFromAlphaToOmega11 ай бұрын
Cubic formula next, please!
@blackpenredpen11 ай бұрын
😆
@benberlowitz638111 ай бұрын
Please do a video where you triple integrate with respect to all 3 im genuinely intrigued what the result would be (is it even possible??)
@seanoneill209811 ай бұрын
You are doing good things !!
@eliteteamkiller3197 ай бұрын
The ending had my rolling.
@cameronskellams567011 ай бұрын
Thats hectic! Love the definitive proof at thr very end that chatgpt clearly has discalculia 😂
@jok200011 ай бұрын
Just ask it to write a symbolic integrator in Python.
@Napsap188 ай бұрын
Love how he doesn’t say everything but says everybody
@StepKar11 ай бұрын
Integrate the qubic formula next
@Protactinium9111 ай бұрын
I’m looking forward to learning hyperbolic tangent!
@pskv2011 ай бұрын
If it is the quadratic formula, why not to use this fact? ∫ x da = xa - ∫ a dx, where ax^2+bx+c=0 -> a=-(bx+c)/x^2 ∫ x da = ax + ∫ (b/x+c/x^2) dx = ax + b ln |x| - c/x + C_1 and substitute x :)
@fengcheng350711 ай бұрын
Yes much easier that way.
@pandabearguy111 ай бұрын
I got the right answer with chatGPT (gpt 4 w/ plugins). Now integrate the cubic formula
@madbrad628211 ай бұрын
That was fun to watch.
@parkerschmitt159411 ай бұрын
Do the triple integral over a b and c
@99chartered11 ай бұрын
Thanks so much, it was fun!
@deldc11 ай бұрын
My eyes are burning! Too many knowlegde! This is insane!!!!
@pwkn8611 ай бұрын
i pay for gpt 4 and its all in how you ask it, so far its done everything you have as well. and was very close to the same answer. AI can and will replace useless professors, as professors are not available to answer students questions at the scale of universities. we NEED ai to help students lean and ask questions when they are lost or FEEL lost. gpt has helped me immensly in my exploration and understanding of calculus. so far, it hasnt been wrong once.
@pradhyumnajadhav913711 ай бұрын
please do triple integration on this next
@parzflash161911 ай бұрын
We got the integral but can you explain the solution graphically as integration with limits gives area and what would the area of this graph be
@acesu578511 ай бұрын
What I've noticed about every math application solver or ai's is that they never used standard forms
@perspicacity8911 ай бұрын
Fantastic video, thank you.
@cheeseparis111 ай бұрын
Thanks, this is what I needed to prove the last Fermat theorem. This may not fit in a KZbin comment actually.
@Saber097 ай бұрын
I am drunk right now but dude this video is absurd, it would never get to my mind this solution! Great! Have a good day :D
@PunyaMalhotra11 ай бұрын
The last integral dU÷U²-b² Is equal to (1÷2b)×ln|(u-b)÷(u+b)| The above result is definitely okay, idk about the one you used(i am nkt saying yours one is incorrect, its just that i never saw that formula😅)
@jd911911 ай бұрын
Who would've guessed that ChatGPT would suck at doing integrals.
@AdaDenali11 ай бұрын
I am struggling to interpret what the area under this curve would represent.
@blatinobear11 ай бұрын
Whenever I see calculus integration and above I’m like, “come on y’all, what are we doing here, let’s go home”
@arnold-pdev11 ай бұрын
Worth noting this function has a domain restricted to the interval 0 =< 4ac =< b^2. The second inequality comes from the discrimant (existence of roots). The first is a statement that either a and c are both negative or both positive. This is weird to me!
@michaelbaum679611 ай бұрын
Nice solution - great👍
@blackpenredpen11 ай бұрын
Thank you!
@vineboomsoundeffect539511 ай бұрын
Why is ∫1/(u^2-b^2)du not equal to 1/(2b) ln|(u-b)/(u+b)|+C in your calculation?
@Jesuisunknown11 ай бұрын
I don't understand a lot but I'm getting interested again to learn the basics of Calculus
@augustodutra38396 ай бұрын
You.should have done a triple integral and integrated with respect to a,b and c all together.
@composerlmythomorphic263511 ай бұрын
I got another (perhaps complicated solution: Suppose the integrand is x, where y=ax^2+bx+c. Then dy/dx=2ax+b, and dy/da=x^2. Hence da/dx=(2ax+b)/x^2. Then integral=int[(x)(2ax+b)/x^2]=int[2a+b/x]=2ax+b lnx +C
@justanotherguy46911 ай бұрын
I love your t-shirt.
@danielsaldivar562211 ай бұрын
Doing math with a smile!
@thec235911 ай бұрын
Completing side-quests
@romanbykov592211 ай бұрын
very interesting video, blackpenredpenbluepen! :)
@blackpenredpen11 ай бұрын
Thank you!!
@Ealron35611 ай бұрын
I love 4 am integration
@yigiteldek11 ай бұрын
i retried asking the integral to gpt couple times and with some hand it eventually integrated the formual exactly as in the video eventually
@jacobblum61811 ай бұрын
Could you do double integration to integrate based of two variables?
@blackpenredpen11 ай бұрын
You can see my 100 integrals part 2. I did about 20 double integrals there toward the end.
@bhupendernayakIITDELHI11 ай бұрын
I don't know in which stage of learning calculus has been taught in other countries we but here in India it is a piece of cake for even High school students 😂😂