Partial Derivatives - Multivariable Calculus

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  • Опубликовано: 18 май 2017
  • This calculus 3 video tutorial explains how to find first order partial derivatives of functions with two and three variables. It provides examples of differentiating functions with respect to x or with respect to y using techniques such as the product rule, quotient rule, and power rule. This video contains plenty of examples and practice problems.
    3D Coordinate System: • Plotting Points In a T...
    3D Distance Formula: • How To Find The Distan...
    Equation of a 3D Sphere: • How To Find The Equati...
    Calculus 3 - Intro to Vectors: • Calculus 3 - Intro To ...
    Calculus 3 - The Dot Product: • Calculus 3 - The Dot P...
    Angle Between Two Vectors: • Finding The Angle Betw...
    Parallel & Orthogonal Vectors: • Are The Two Vectors Pa...
    Direction Cosines and Vectors: • Calculus 3 - Direction...
    Calculus 3 - Vector Projections: • Calculus 3 - Vector Pr...
    Cross Product of 2 Vectors: • Cross Product of Two V...
    _______________________________________________________________________________________
    Area - Vector Cross Product: • Area of a Parallelogra...
    Triple Scalar Product: • Volume of a Parallelep...
    Vector Equations of Lines: • How To Find The Vector...
    The Equation of a Plane: • How To Find The Equati...
    Planar Equation - 3 Points: • How To Find The Equati...
    Lines & Planes - Intersection: • How To Find The Point ...
    Angle Between Two Planes: • How To Find The Angle ...
    Distance Between Point and Plane: • How To Find The Distan...
    Chain Rule - Partial Derivatives: • Chain Rule With Partia...
    Implicit Partial Differentiation: • Implicit Differentiati...
    _______________________________________________________________________________________
    Directional Derivatives: • How To Find The Direct...
    Limits of Multivariable Functions: • Limits of Multivariabl...
    Double Integrals: • Double Integrals
    Local Extrema & Critical Points: • Local Extrema, Critica...
    Absolute Extrema - Max & Min: • Absolute Maximum and M...
    Lagrange Multipliers: • Lagrange Multipliers
    Triple Integrals: • Triple Integrals - Cal...
    2nd Order - Differential Equations: • Second Order Linear Di...
    Undetermined Coefficients: • Method of Undetermined...
    Variation of Parameters: • Variation of Parameter...

Комментарии • 1 тыс.

  • @TheOrganicChemistryTutor
    @TheOrganicChemistryTutor  6 месяцев назад +17

    Final Exams and Video Playlists: www.video-tutor.net/
    Full-Length Videos & Worksheets: www.patreon.com/MathScienceTutor/collections

  • @Wasoska
    @Wasoska 4 года назад +4416

    How many engineer students here ?

    • @GiuResta
      @GiuResta 4 года назад +112

      no one cares about geology students :(

    • @Wasoska
      @Wasoska 4 года назад +80

      @@GiuResta in my college this is calculus 3 and for what I know, only engineers have to take calculus 3.

    • @dudeisawesome7202
      @dudeisawesome7202 4 года назад +35

      @@Wasoska Same, although there are some computer science students in my class lol

    • @danialasmady2953
      @danialasmady2953 4 года назад +16

      @@Wasoska In my university, geology students have to take these as well :(

    • @josephsweitzer5471
      @josephsweitzer5471 4 года назад +23

      Lourdes Rodz I have to do this for chemistry and physics, like WTF. I want to die.

  • @erikmachado4489
    @erikmachado4489 3 года назад +1210

    What truly makes this channel great is he takes his time to explain a prior concept, so that we can fully understand what's going on. Most channels just say something like "I assume you know how to do this part" and proceeds to solve the problem forcing students to go back and remember something they haven't seen in a bit when literally a simple review can come a long way.

    • @undefined.infinity3106
      @undefined.infinity3106 2 года назад +51

      that way he shows every derivative part by part, irl any teacher never do that for their students.

    • @devalon8568
      @devalon8568 2 года назад +5

      He sometimes does this too, though. But yea better than other teachers.

    • @blurryface616
      @blurryface616 2 года назад +2

      It needs patience to do that and most teachers don't have it.

    • @bruceleroyalty
      @bruceleroyalty Год назад +2

      This channel has saved me in the physics and calculus series both 🙏

    • @anyegodlove3546
      @anyegodlove3546 Год назад

      it's well simplified

  • @chemthingz7467
    @chemthingz7467 5 лет назад +1171

    To express my gratitude, I watched all of the ads that appeared

    • @ronallan8680
      @ronallan8680 5 лет назад +64

      You need to click them. if you click thats when they get some revenue. I always click his ads

    • @markcross109
      @markcross109 5 лет назад +10

      Salamat mahal ko.

    • @pandamamoosshhhshek9512
      @pandamamoosshhhshek9512 5 лет назад +8

      Omg same. He's really a savior.

    • @lynns4122
      @lynns4122 4 года назад +3

      Nice

    • @wrjazziel
      @wrjazziel 4 года назад +1

      Good one!

  • @sophiaonyoutube
    @sophiaonyoutube Год назад +337

    It's 2022 and this guy's still saving me from failing my exams. Thank you so much, you are a hero!

    • @SharifKhan-tz9pn
      @SharifKhan-tz9pn Год назад +3

      same to you.

    • @ritik7906
      @ritik7906 9 месяцев назад

      true 5 days before exam

    • @MichelleHugli
      @MichelleHugli 7 месяцев назад

      Day of the test!!!!

    • @CharlesMosesEmeruka-kz9xt
      @CharlesMosesEmeruka-kz9xt 6 месяцев назад +1

      He is more than just a hero. He is just a God given Gift to humanity's education. Wow! I love the way he passionately explains the concepts.

    • @random1ndividual
      @random1ndividual 6 месяцев назад +2

      almost 2024 for me lmao

  • @joshp3446
    @joshp3446 4 года назад +368

    this guys is absolutely amazing, it seems like he knows everything, and can translate it into dumb for the rest of us. Thank You O.C.T!

    • @Samdala.
      @Samdala. 4 года назад +10

      Josh P 🤣🤣🤣 thank you for the comment brodah

  • @juleslcj1396
    @juleslcj1396 Год назад +130

    Again thank you for your great work. You've literally taught me calculus 1,2 and now three. You saved me in my high school chemistry and physics classes especially during the covid lockdown. My high school grades in chemistry and physics are literally because of you. Basically you're the reason why I was able to reach the qualification for an engineering degree and you're still helping me survive the degree. Thank you man! Honestly I owe you.

  • @peachfreude
    @peachfreude 5 лет назад +218

    I love that you giving us many examples question and then explain in details how to solve the questions. You're such a good teacher. Thank you so much!

  • @samuelalves905
    @samuelalves905 Год назад +36

    I cannot express how much you help me through school. It’s not just my grades you’re helping me improve, but my mental state. You quite literally save lives.

  • @sentinel_1752
    @sentinel_1752 4 года назад +86

    You are the greatest teacher I’ve ever had and I’ve never even seen you.
    Thanks for sharing the education to all of us in need.

  • @angelahuynh5074
    @angelahuynh5074 Год назад +36

    You deserve like half of my degree because the number of classes you've pulled me through so far as a college junior is unreal

  • @austinmorris455
    @austinmorris455 2 года назад +14

    4 years later and you’re saving me the night before my calculus final. thank you bro❤️

  • @hughpearse
    @hughpearse 6 лет назад +14

    Best Partial Derivatives video so far on youtube, great job!

  • @juleslcj1396
    @juleslcj1396 Год назад +13

    You have done what my lecturer could not accomplish in his three hour lecture. Thank you so much, you're a life saver.

    • @michaelbujaki2462
      @michaelbujaki2462 Год назад

      And he can do it in 60, 48, 40, or 30 minutes depending on how fast you can listen.

  • @mphomokhwebane5696
    @mphomokhwebane5696 3 года назад +9

    this person is a master,he knows almost everything academically

  • @kerialyscoss1289
    @kerialyscoss1289 6 лет назад +21

    LOVE LOVE THIS VIDEO. THANK YOU SO MUCH, YOU EXPLAINED EVERYTHING SO DETAILED AND EVEN THE BASIC DERIVATIVES FROM CALC 1. THANK YOU SO MUCH!!!!!

  • @LustrousKingMax
    @LustrousKingMax 4 года назад +14

    I don't think you understand how great you are sir. You are legendary.. I can't imagine varsity without your youtube videos!

  • @wardahassan15
    @wardahassan15 4 года назад +6

    istg your channel is the reason im passing calculus. Bless your soul dude

  • @jasminscheven7732
    @jasminscheven7732 Год назад +6

    This really laid down the basics I needed for my Computational Finance degree. Would have been lost without it. Much appreciated!

  • @famwang7090
    @famwang7090 3 года назад +5

    VERY HELPFUL!!! For a first time learner, this video was extremely clear and direct to the point.

  • @stevenwhitesell1804
    @stevenwhitesell1804 3 года назад +15

    The pointer on when to use and not to use the product rule @ 21:10 is excellent. Thank you!

    • @dannynufer1296
      @dannynufer1296 Год назад

      Honestly!! I've been fucking up so much through using product rule when I didn't need to - That little tip, though it seems obvious now he's explained it, has helped me no end haha.

  • @lj54
    @lj54 7 лет назад +32

    Thank you! I've been following your video's since Calculus I. I'm now in Calculus III, so my excitement upon seeing this video was pretty amazing. Thanks!
    I'm taking Differential Equations next semester and it would GREAT if you could add some of those videos!
    Thank you!

  • @Czar684
    @Czar684 3 месяца назад +1

    You were one of my pillars of support that helped me get through college and earn a degree of Bachelor of Science in Civil Engineering.

  • @nkhjk3457
    @nkhjk3457 4 года назад +44

    forgot to wake up for online class , this guy saved my ass XD

  • @sia7001
    @sia7001 6 лет назад +5

    THIS VIDEO SAVED MY LIFE! THANK YOU SO MUCH! BLESS YOUR SOUL :'D

  • @m4trixbee477
    @m4trixbee477 2 года назад +3

    this is the only 1 hour educational video that makes me want to listen and learn

  • @user-br3mg5gt3g
    @user-br3mg5gt3g 4 месяца назад

    You have no idea how much you've helped me with your videos through the years! Even in the moments in which math is hard and misunderstandable you come and the bright comes with you! Be blessed!
    Greetings from Bulgaria and Technical University of Sofia!

  • @ordinaryus5753
    @ordinaryus5753 2 месяца назад

    I've got calculus exam in 2 days, doing final reviews, massive thanks man. You are genuinely a savior

  • @brindaprothia9973
    @brindaprothia9973 4 года назад +28

    When I don't understand my calc prof's lecture, I don't stress because I know that I can find your videos and understand everything! Thank you so much for your excellent way of teaching! :)

    • @TrinidaddyGdom
      @TrinidaddyGdom Год назад +4

      I missed every class before my exam tomorrow and im feeling confident tbh lol

  • @shathyanaraynanb9506
    @shathyanaraynanb9506 4 года назад +4

    Thank you so much really helped me a lot with the exams approaching ...thanks once again

  • @celsolompot6556
    @celsolompot6556 3 года назад

    I didn't skip all the ads. That's the only way to thank you! Thank you so much!

  • @Aim4sixmeals
    @Aim4sixmeals 5 лет назад +1

    Better than other channels showing how to do these things. This guy is straight to the point no bullshit/crap whatsoever.

  • @centainejoseph337
    @centainejoseph337 5 лет назад +120

    Can you do a video on calculus 3 topics that fall under partial derivatives such as
    1. Limits and Continuity in Higher Dimension
    2Directional Derivatives and Gradient Vectors
    3. Tangent Planes and Differentials
    4.Extreme Values and Saddle Points

    • @wrjazziel
      @wrjazziel 4 года назад +19

      Do you want a pony too? ....
      ... JK man LOL

    • @ahmet8266
      @ahmet8266 4 года назад +6

      @@wrjazziel haha very funny

    • @deedeeworksitout186
      @deedeeworksitout186 4 года назад +9

      Check out Math TV with Professor V, there’s videos about those topics!

  • @adonaig.446
    @adonaig.446 7 лет назад +33

    Thanks a bunch. I find all of your videos very helpful. Some people just have the gift to teach.
    Could you please consider a series on quantitative chemical analysis?

  • @daves333
    @daves333 3 года назад

    This video is perfect. You made sense of something that wasn't clicking with me at all. Thank you so much.

  • @kalumakalongojnr4346
    @kalumakalongojnr4346 2 года назад

    This man sounds calm,kind and humble,,,.good work sir

  • @mansfieldclayn7994
    @mansfieldclayn7994 Год назад +5

    This man is great, I love this man’s body, I love this man’s body of work.

  • @peteporcelli7829
    @peteporcelli7829 5 лет назад +4

    my first semester i lied about taking calc when i enrolled in physics so i had to watch your video on derivatives.
    now im taking meteorological measurement and lied about takin calc 2 so yeah youre the bomb bro.

  • @jacobschmitt7033
    @jacobschmitt7033 3 года назад +1

    you deserve every penny (and more) that you make from these videos.

  • @crazycuban420
    @crazycuban420 5 лет назад

    The clarity of your videos is beautiful.

  • @billionaire7781
    @billionaire7781 6 лет назад +353

    Way better than khan academy

    • @ArsArcanum
      @ArsArcanum 5 лет назад +46

      khan academy sucks. they over explain stuff to the point where you're lost, then ask you to solve. Khan academy only works for basic match, etc.

    • @tinambalienhle7691
      @tinambalienhle7691 5 лет назад +15

      waaaaaaaaaaaaaaaay better..

    • @atp4223
      @atp4223 4 года назад +1

      khan academy belongs to the trash

    • @rjarcana4747
      @rjarcana4747 4 года назад +60

      @@atp4223 hey dont talk like that. atleast theyre providing free teaching for :)

    • @e.b.1279
      @e.b.1279 4 года назад +20

      While I find The Organic Chemistry Tutor FAR SUPERIOR to Khan Academy, I would never say it's outright bad. In fact, I would say that Khan Academy "paved the way" for (better) future online STEM tutors!! I wouldn't be surprised if even O.C.T. in younger days spent even some small amount of time with Khan Academy content.
      However, this is CLEARLY far, far superior!!

  • @hassanabdulla3824
    @hassanabdulla3824 7 лет назад +75

    I hope to you explain this topics:
    1-static .
    2-calculus 2 , 3.
    3-dynamics.
    4-linear algebra.
    5-material science.
    6-fundamentals of physics and bulk properties of matter.
    7-engineering thermodynamics .

    • @Vermouth.4869
      @Vermouth.4869 5 лет назад

      hassan abdulla yes please

    • @louishennick12
      @louishennick12 5 лет назад +5

      Engineering student huh lol

    • @lynns4122
      @lynns4122 4 года назад +7

      3blue1brown has a pretty cool linear algebra series.

    • @maija6099
      @maija6099 4 года назад +1

      Metallurgical student huh?

    • @loona8398
      @loona8398 4 года назад +1

      computer science ?

  • @Australis_
    @Australis_ 2 года назад +1

    This video is gold, thanks so much for taking the time to do it!

  • @Youtuber-xc8vl
    @Youtuber-xc8vl 2 года назад +1

    this honestly is one of your best works. thank you so much

  • @conamore7887
    @conamore7887 3 года назад +7

    I never heard of partial derivative before but after watching this tutorial, I've got my basic concepts.شكرا يا استاذ

    • @dkke825
      @dkke825 Год назад

      ruclips.net/video/Zuh5EiW0o7w/видео.html

  • @djamelhimani8012
    @djamelhimani8012 4 года назад +4

    thank u so much sir , u helped me a lot and i wish inchallah what i have learned will help me in the exam that i will pass this next week

  • @zurzakne-etra7069
    @zurzakne-etra7069 3 года назад +1

    I've watched this 3 times now. So very helpful!

  • @kailyn3600
    @kailyn3600 5 лет назад +1

    I would just like to say thank you....your videos helped me pass college calc 1 and 2.

  • @asaddtariq
    @asaddtariq 4 года назад +6

    hey Bud, i love you so so much, that i can't explain to you. i got 2.7 gpa in just a week, because of you thanks

  • @Shaylicious___
    @Shaylicious___ 6 лет назад +31

    Thank you so much, I can't even explain how much you've helped me 😩😩♥️♥️🙌🏽✅🤸🏽‍♂️👌🏽

  • @user-bd4wx7pp9q
    @user-bd4wx7pp9q 4 года назад +1

    ربنا يكرم حضرتك ياااارب تستاهل كل خير علي مجهودك بالتوفيق دايما يااارب ♥️♥️🌷

  • @timwigglesworth6873
    @timwigglesworth6873 4 года назад +1

    watched it in 10 mins could have been shorter but thank god someone finally talked about fxx and fxy and all that cheers chief

  • @abdulnazary1501
    @abdulnazary1501 6 лет назад +12

    TOCT, If you ever get a chance it would be a big help if you can make videos covering the unit calculus 3 (multivariable calculus). There are other videos on this subject, but the instructors do not teach the way you do. Thanks!

  • @jojo-lj7tc
    @jojo-lj7tc 5 лет назад +3

    Thank you so much for helping me please can you make a video on the limit of multivariable

  • @guruprasath8548
    @guruprasath8548 3 года назад +1

    Really appreciate the effort you took to teach us this.

  • @ahmediu9984
    @ahmediu9984 Год назад +2

    I am from Sudan and i just want to tell you that you are the only reason i am getting excellent grades and i also love math because of you beautiful way of teaching and making us understand the hardest equations using the simplest terms and detailed explanation thank you very much ❤️❤️❤️❤️❤️❤️❤️

  • @NAWAFf15
    @NAWAFf15 6 лет назад +4

    انا اشهد بالله انك اسطوره

  • @LongLiveMasculinity
    @LongLiveMasculinity 5 лет назад +3

    on 12:05 you could also apply laws of logarithm whereby z=lnx^2 -lnY

  • @hi77777
    @hi77777 3 месяца назад

    You are an angel, you deserve so many blessings in your life!

  • @CharlesMosesEmeruka-kz9xt
    @CharlesMosesEmeruka-kz9xt 6 месяцев назад

    Oh my God! You made partial derivatives so easy for me to understand. You are simply AMAZING!! WOW!!!

  • @rohit3265
    @rohit3265 7 лет назад +8

    your videos are awesome but can you upload videos for calculus used in physics

  • @catcen9631
    @catcen9631 2 года назад +11

    It's a great video! I only have one question left... How do I know when the constant becomes 0 or stays the same, when we do differentiate it?

    • @realndu
      @realndu 2 года назад +7

      If we're differentiating with respect to x, y is considered a constant and vice versa. If you're differentiating with respect to one of the variables and the other variable is a stand alone, it will become 0. If the other variable is the product to the variable being differentiated respectively, then its ignored and not made to be 0. Hope this helps :)

    • @Erik-ed8ml
      @Erik-ed8ml Год назад

      @@realndu this comment saved my life thank you

    • @realndu
      @realndu Год назад

      @@Erik-ed8ml You're welcome!!

  • @jasminewashington8076
    @jasminewashington8076 4 года назад +2

    Thank you so much for making this video! You are a life saver.

  • @aly7514
    @aly7514 2 года назад +2

    thank you so much for this!

  • @eray7232
    @eray7232 Год назад +4

    even x and y get more respect than me

  • @flamingstar1604
    @flamingstar1604 5 лет назад +7

    I need a lesson about maximum and minimum for function of two variables

  • @austinmakachola2516
    @austinmakachola2516 6 лет назад +1

    Thank you very much. You've made my work easier.

  • @raphaelling0663
    @raphaelling0663 3 года назад +2

    Thanks. Easy to understand and able to get an A+ for this subject.

  • @Golden_348
    @Golden_348 7 лет назад +4

    Please show me the mechanism of baking process

  • @j.padilla309
    @j.padilla309 5 лет назад +15

    Hi,
    Why some constant, you leave them as it is, and some you derive them to 0. I am confused. Thanks.

    • @hideyoshi4794
      @hideyoshi4794 5 лет назад +13

      You leave them when the constants are next to each other without a + in-between , if it's alone you derive it to 0. I hope you understand and Sorry for the late reply :)

    • @bigmann8932
      @bigmann8932 4 года назад

      Catch The Cougar thank you

    • @darrenvillaroman7736
      @darrenvillaroman7736 3 года назад

      @@hideyoshi4794 thanks a lot

    • @hideyoshi4794
      @hideyoshi4794 3 года назад +2

      @@darrenvillaroman7736 im glad that i helped sir

  • @IvyANguyen
    @IvyANguyen 6 лет назад +1

    Thanks for this on how partial derivatives work. This was a big help.

  • @tedle2948
    @tedle2948 3 месяца назад

    I can't imagine there is a video like this, thank you sir

  • @Pavat__Ke
    @Pavat__Ke 5 лет назад +3

    2 years later and the video is still super useful..

  • @abrarmahi
    @abrarmahi 3 года назад +6

    where my comp sci students at

  • @kaviamedi8079
    @kaviamedi8079 Год назад +1

    This video is really helpful in understanding the partial derivatives. thank you so much!

  • @youshaarif4253
    @youshaarif4253 11 месяцев назад

    Thank you, that exactly explained what I was looking for. A lot of love an respect from Japan

  • @somalilanderman6429
    @somalilanderman6429 5 лет назад +3

    For the first example
    When we were finding fy?
    What is the reason that 7 x.squered become =0??????

    • @markcross109
      @markcross109 5 лет назад +1

      There's no "y" variable in that term, and we know that differentiation can extend until you reach zero from y' to y^infinity.

    • @somalilanderman6429
      @somalilanderman6429 5 лет назад

      @@markcross109
      Thanks alot my freind💔

    • @markcross109
      @markcross109 5 лет назад +1

      @@somalilanderman6429 welcome my friend.

  • @eggieeeeeee
    @eggieeeeeee 4 года назад +9

    People, I'm only 14 years old, but I am l learning Multivariable Calculus! Thanks to this guy.

  • @NiiAryee
    @NiiAryee 2 года назад

    Excellent as always, in simple, plain English.

  • @tanimhasan6089
    @tanimhasan6089 2 года назад

    You shouldn't thank us for watching rather we should say thank you for saving us each and everytime!!! You're the best Organic Tutor......

    • @dkke825
      @dkke825 Год назад

      ruclips.net/video/Zuh5EiW0o7w/видео.html

  • @yamkelaniking8380
    @yamkelaniking8380 3 года назад +3

    dude lets just get married and have Organic chemistry Tutor babies already

  • @miribear3809
    @miribear3809 Год назад +11

    How many Econ students here?

  • @EpicVisionary.
    @EpicVisionary. Месяц назад +1

    jzakAllah sir for your efforts. Your way of teaching is just amazing

  • @namehere630
    @namehere630 3 года назад

    my gratitude is beyond words. char

  • @everchann
    @everchann 11 месяцев назад +24

    how many computer science students here?

  • @haiderrajput4315
    @haiderrajput4315 5 лет назад +2

    Really good effort.. I appreciate your efforts for helping other by your knowledge! 👍👍

  • @pog9238
    @pog9238 Год назад +1

    YOU ARE A LEGEND, YOU ARE THE BEST OF THE BEST

  • @chisomnwodo2905
    @chisomnwodo2905 4 месяца назад

    GOD BLESS YOU FOR SHARING YOUR WISDOM . You've literally helped and saved so many students from failing . You're a life saver !

  • @BERLINFASHIONFOU
    @BERLINFASHIONFOU 5 лет назад +2

    This video is awesome! Thank you so so much! Love your account

  • @shafaqueiqbal9728
    @shafaqueiqbal9728 8 месяцев назад

    The video was just amazing. It clears all my doubts. Thank you so much 👍👍

  • @war_correspondent
    @war_correspondent 3 года назад +1

    Thank you very much sir. You are helping me for my calculus test tomorrow

  • @peerapongwattanthaweesin8304
    @peerapongwattanthaweesin8304 Год назад +2

    Absolutely helpful, thanks my guy.

  • @the_kid777
    @the_kid777 8 месяцев назад

    These days, I was tackling math without the help of your videos. But today, after crying for the whole day because I couldn't understand a single thing, I finally realized that doing math without your videos was an impossible attempt. Thank goodness I made the decision to search up for your video on RUclips.

  • @iharshgarg
    @iharshgarg 3 года назад +1

    khan academy and this organic chem tutor.. probably the best content u can find on utube.. so stop paying ur tuition teacher and start support him ❤

  • @ndereratjijahura2048
    @ndereratjijahura2048 2 года назад

    Thank you so much sir. All your presentations are clear. You deserve prize from me.

  • @rimshy
    @rimshy Год назад

    This lecture solved all of my confusions about partial derivatives

  • @scr34m_piu
    @scr34m_piu 5 лет назад +2

    Damn, this 1 hour is saving my semester bro. Keep up ur good work!!!!!!!

  • @Name-zm5lk
    @Name-zm5lk 7 месяцев назад

    YOU ARE MY SAVIOR. Thank u soooo much for the video

  • @solomon5708
    @solomon5708 2 месяца назад

    youre saving my grade, thank you so much!!!

  • @makuwathembi1521
    @makuwathembi1521 5 лет назад +1

    you really know how to explain...big up to you