What is a Derivative? Deriving the Power Rule

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  • Опубликовано: 29 сен 2024
  • After discussing differentiation at great length, it is time to connect this concept with the act of taking the derivative of a function. In actuality these mean the same thing, but using the power rule to take the derivative of a function is actually much simpler than all that business with limits and tangent lines! But it is important to know what the power rule is and where it comes from, so let's derive it together. This way, when you take the derivatives of functions, you'll know exactly what is really happening.
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Комментарии • 156

  • @TheKarstrasse
    @TheKarstrasse 4 года назад +169

    I was never taught like this in school because even our teachers didn't know all these. We directly started with dy/dx without understanding what they mean.
    Thanks professor Dave. I'm a software application design architect since a couple years and still watching this now

    • @yamanydadwaria
      @yamanydadwaria 3 года назад +5

      Absolutely ! right. 🤞🏻

    • @MP-cv6if
      @MP-cv6if 3 года назад +7

      Same. My teacher did know all this, but he assumed since we are the top students we must already know things like these and skipped over to questions. I was literally in tears at the end of our second class.

    • @batiandanielliu3598
      @batiandanielliu3598 Год назад +1

      I'm also a software application design architect, well it's fun to watch this video to go through the logic of human mind

    • @The-Cosmos
      @The-Cosmos 8 месяцев назад

      Schools are stupid

  • @metayoutubeacademy
    @metayoutubeacademy 2 года назад

    I just cried of joy, how easy you made my life, Love You Professor Dave

  • @mobiusII
    @mobiusII 2 года назад +1

    Great video! I wish my calculus teacher had spent some time demonstrating what the 'h' represented in the more formal method of finding derivatives. Now that I understand that, the power rule is more meaningful to me, and not some neat mathematical trick. Thank you! Also, I'm a science teacher and I show your videos on Newton's Laws of Motion to my classes every year.

  • @tbax4259
    @tbax4259 5 лет назад +3

    Very helpful videos, Prof. Dave!

  • @ahappyimago
    @ahappyimago 6 лет назад

    How did you get 0 from f(x+h)-f(x) when x =5? Shouldn't it be 5+h-5/h?

    • @ProfessorDaveExplains
      @ProfessorDaveExplains  6 лет назад +2

      but if f(x) = 5, then no matter what h is, the function is always equal to 5.

    • @ahappyimago
      @ahappyimago 6 лет назад

      Professor Dave Explains Thanks, Dave. You are the man. I'm going to donate to your Patreon when I get back to LA from my business trip. Your channel is very precious and with the right exposure it will definitely become one of the premier educational channels.

    • @ProfessorDaveExplains
      @ProfessorDaveExplains  6 лет назад

      woohoo! thanks in advance for the contribution :)

    • @ahappyimago
      @ahappyimago 6 лет назад

      Alright bro, I'm a Mega Professor Pal.

    • @ProfessorDaveExplains
      @ProfessorDaveExplains  6 лет назад

      awesome!!! you are a true hero!

  • @sujitsuram5075
    @sujitsuram5075 5 лет назад +182

    I don't know why but i like it when you give pauses between statements, helps me sync in a bit. Appreciate the work you have done so far.

    • @camhux
      @camhux 7 месяцев назад +2

      I noticed this too, just a little second to acknowledge that the audience is absorbing something. instinct of a great educator

  • @bd_ART
    @bd_ART 5 лет назад +31

    Just As always , the best courses in the world

  • @tasnimulsarwar9189
    @tasnimulsarwar9189 3 года назад +20

    Never, like never in my life have I ever been taught the preliminary of the derivative like this way. Never thought I would understand how we get to the power rule.
    Thank you so much for this. This and your awesome and kindhearted work will always remain in my heart till the end of my days.
    Once again, thank you so much.

  • @abhisheksingh-qv5el
    @abhisheksingh-qv5el 5 лет назад +28

    After seeing this I think in India only focus is on how to put values on formula... many teachers actually don't know why they are doing so...
    You are doing great job..keep it up sir

    • @malwatmalwat8447
      @malwatmalwat8447 4 года назад

      👍👍👍👌👌

    • @TheKarstrasse
      @TheKarstrasse 4 года назад +1

      Exactly

    • @gnb-kd2it
      @gnb-kd2it 3 месяца назад

      what rank did you get in jee you did not qualify am i right

    • @slanderking69
      @slanderking69 24 дня назад

      ​​@@gnb-kd2it Jee is a pointless examination. The only reason it is that hard is because India has a gaint population. By understanding these fundamentals behind maths we can see how beautiful maths is.

  • @kripashankarshukla4073
    @kripashankarshukla4073 5 лет назад +27

    Your explanation is of god level this is definitely the best calculus tutorial in the world I am very lucky to get it

  • @furyplayer4836
    @furyplayer4836 4 года назад +8

    this professor just took the entire 1 week of calculus in just 10minutes video... Godbless you sir for making these videos

    • @The-Cosmos
      @The-Cosmos 7 месяцев назад +1

      Atheism bro.

  • @kafkaesquegoku
    @kafkaesquegoku 5 месяцев назад +3

    PROFESSOR DAVE THIS VIDEO LITERALLY CHANGED THE WAY I VIEW CALCULUS THANK U SO MUCH 😭😭😭 seeing the difference quotient being visualized changed everything!!!!!

  • @igotapochahontas
    @igotapochahontas 5 лет назад +7

    Best explaination for WHY the derivitive of a constant is zero ive ever seen. This is quickly becoming one of my favorite calculus channels. Have you considered doing some practical, easy, applied calculus videos? Everyone seems to dodge that subject. Id like to see some simple applications of calculus without having to bore myself to sleep with endless strings of algebra.

  • @Bilbus7
    @Bilbus7 6 лет назад +11

    Math is neat.

  • @randoshmuckarias1296
    @randoshmuckarias1296 3 года назад +4

    Thanks a lot Professor Dave! you really helped me understand it better. Since in our schools they simply make us memorize that the derivative of a number is 0 and the power rule and so on directly.

  • @lhaviland8602
    @lhaviland8602 4 года назад +3

    I feel extremely lucky lol. My teacher explained it pretty much exactly like this. Seems that isn't the case everywhere sadly.

  • @tasnimulsarwar9189
    @tasnimulsarwar9189 3 года назад +2

    Never, like never in my life have I ever been taught the preliminary of the derivative like this way. Never thought I would understand how we get to the power rule.
    Thank you so much for this. This and your awesome and kindhearted work will always remain in my heart till the end of my days.
    Once again, thank you so much.

  • @gr5565
    @gr5565 9 месяцев назад +2

    I have always appreciated how beautifully one can understand physics when you derive the equations instead of only memorizing them. One can understand the true meaning of those equations. You just did that for me with derivatives. I have watched countless videos, many very good, but yours is the first one I remember explaining where the power rule comes from. Just beautiful!

  • @DaReelGamerz
    @DaReelGamerz 8 месяцев назад +1

    Teachers these days should learn from Prof. Dave. I've never seen a clearer explanation of derivatives.
    Finally understood calculus.
    THX❤

  • @dumpstercat2229
    @dumpstercat2229 9 дней назад +1

    Thanks bunch! This really helped me a lot

  • @FlydingVent9296
    @FlydingVent9296 Год назад +1

    The proof for why it is nx^(n-1) wasn't shown, you just assumed in the video by conjecture by observing a pattern, that is not enough because what may look like a pattern might not be a pattern at all. I'm really enjoying these well thought and well made videos and the context and visual tools provided, but omitting proofs end up giving the students the false idea that they understood the why of something

    • @melancall5960
      @melancall5960 11 месяцев назад

      This is the introductory concept for Derivatives, the proof for the Power Rule is in the calculation itself. nx^n-1 remains consistent over any given positive power integer.
      x^2 = 2x, x^3 = 3x^2 is fun, but the rule applies for even larger values such as 34x^12 = 34•12x^11 -> 408x^11
      Sure it may not work as smoothly for calculations with more steps, but again this is introductory

  • @vinitsunita
    @vinitsunita 3 года назад +2

    I am 32 years old and first time in my life I actually understood what power rule is and what is the application of differentiation. This is amazing video

  • @kavyahegade6477
    @kavyahegade6477 4 года назад +1

    No person can beat him or even compete with him

  • @greenzie7099
    @greenzie7099 Год назад +1

    Professor Dave can make me watch a lesson about calculus even after I've passed

  • @kamidelstudio
    @kamidelstudio 5 лет назад +13

    How does someone make calculus entertaining?!? Thank you for your videos

  • @lananhtran1124
    @lananhtran1124 3 года назад +1

    All these years I always thought Calculus is something really big and complicated but turns out it's just a fancier Algebra. And again, thank you, this is the best Calculus tutorial I've ever had.

  • @FlorianEagox
    @FlorianEagox 5 лет назад +2

    Great vid, learned lots.
    I swear everyone has a patreon in a post-adpocolyptic world.

  • @schifoso
    @schifoso 6 лет назад +4

    An excellent explanation of how the power rule is derived.

    • @kjl3080
      @kjl3080 3 года назад

      > derived
      I see what you did there

  • @balayogam
    @balayogam 5 лет назад +2

    wonderful lectures. Thank you very much.

  • @non-applicable.
    @non-applicable. 6 месяцев назад

    This is such a great explanation. Thank you so much!👏👏👏

  • @shariffahpateyahazvi8718
    @shariffahpateyahazvi8718 5 лет назад +2

    my first time watching your maths videos and i was geared to work out comprehension question sine am used to watching your physics videos

  • @natu.d9546
    @natu.d9546 3 месяца назад

    I remember learning about these, but I think our teacher left out this giant stuff and just told us if f(x) = mx^n then to find the first derivative, it will look like this f'(x) = n[mx^n-1] I didn't even know it was called the "power rule" tnx

  • @SibruOfficial
    @SibruOfficial 6 месяцев назад

    Thank you, math/chemistry/physics/biology jesus!

  • @Daniel___e4e7
    @Daniel___e4e7 2 месяца назад

    An inside look into the future: exclusive interview with Binance’s CEO

  • @raghupathik7563
    @raghupathik7563 6 лет назад +1

    professor please check your mail, i am seeking answer for that qus

  • @Jesuisunknown
    @Jesuisunknown 7 месяцев назад

    3:42 Isn't h→a instead of h →0 in the formula for the tangent line?
    Of course the distance become O

  • @gnb-kd2it
    @gnb-kd2it 3 месяца назад

    i came here for derivation of power rule and you just taught things that i already knew

  • @Kiky_MedPhysicist
    @Kiky_MedPhysicist Месяц назад

    Thank you sir for your dedication and for making this free!

  • @Ruth-v8Ruth_7v7d
    @Ruth-v8Ruth_7v7d 2 месяца назад

    Here come the details and expected actions for your refund

  • @amyrkamal
    @amyrkamal 5 месяцев назад

    he is not explaining maths. he is developing maths nerds 😂

  • @rogerbogh3884
    @rogerbogh3884 Год назад

    Man, my instructors taught me the 'Power Rule' without teaching me how/why it 'worked'.
    Just math, not pattern matching. I was never confident in Calculus because it was not taught from the basics.

  • @wada-wada
    @wada-wada 3 года назад +1

    5:23 why is h in denominator not 0 as h approaches 0
    and we'll get 0/0?

    • @carultch
      @carultch 2 года назад

      Because we aren't actually evaluating the fraction specifically at h=0. The fraction has a hole (as in a removable singularity) at h=0, where both the numerator and denominator equal zero together. We are interested in what the function does, if that hole gets removed, and as the function approaches that hole.

  • @KxGXRish_Gaming
    @KxGXRish_Gaming Год назад

    You are a great teacher please keep creating content... im from india and trying for iit jee exams and your explanation helped me a lot.....it enhanced my basic conceps.....i would say that my proffessors dont even know this much about calculus...thanks to you

  • @Roman_CK
    @Roman_CK 9 месяцев назад

    This video really helped me understand a crucial bit of calculus.

  • @wendykurniawan3932
    @wendykurniawan3932 Год назад

    thank you so much for explaining this. The best explanation of derivatives. All these years and at last someone can explain the logic behind derivatives. My teacher back then only gave me the formula without explaining further.

  • @ronin6158
    @ronin6158 Год назад

    coming back to this series to refresh ideas on how to teach a certain young person calc. Pausing 3:30 is a true boom-shaka money moment.
    also, from 6:03, 'hiding' the *potential* zero's in a variable until the divisor h cancels *then* letting top go to 0, is also an algebra trick fairly critical to understanding.

  • @kaanbakacak9977
    @kaanbakacak9977 Год назад

    I've just discovered you. I hope you will help me to pass my course. Btw your voice and style of speak is awesome. Already fell in love w you

  • @Sakshi-xo7jr
    @Sakshi-xo7jr 3 года назад

    I have a question, at 2:55 u said that the point x approaches point a and so does it eventually meets the latter as you have showed it graphically? And if so wouldn't the values of both point a and x will equal then? And if both points r equal, slope calculated by using x and a will be zero. This shouldn't happen right?

  • @heaven3706
    @heaven3706 4 года назад +1

    You are such a good teacher! Thank you!!

  • @mhadf11
    @mhadf11 9 месяцев назад

    God bless you man. Sincerely

  • @cam2891
    @cam2891 7 месяцев назад

    7.59 where did the 3x go? i would appreciate if someone answers because I don't have time :(

    • @sangamkatuwal9437
      @sangamkatuwal9437 7 месяцев назад

      Its elementary maths
      Value of h is zero so 3x times h is also zero

  • @srirampatnaik9164
    @srirampatnaik9164 4 года назад

    Do you have any recommended worksheet or question bank? Please consider

  • @benbb1758
    @benbb1758 3 года назад

    me learning calculus a year earlier than i was going to so i can understand quantum mechanics: stonks

  • @gokucrafter9456
    @gokucrafter9456 2 года назад

    You ARE one of BEST mentors that I have ever seen in my entire life. Even if I can ask questions with my teacher in class, I can't understand the definition to such a great degree, regardless of the fact that English is not my mother tongue.

  • @codexcodexcodex
    @codexcodexcodex 2 года назад +2

    I'm just 14 yrs old but I now understand calculus (differentiation), thanks to this channel.👍👍👍

  • @josephconder9074
    @josephconder9074 5 лет назад

    At least with regard to algebraic functions, It seems to me that differentiation is a "stripping down" of a function while integration is a "restoration" of it.

  • @WandileBrynt-dk8py
    @WandileBrynt-dk8py 3 дня назад

    am just here for the intro

  • @amyrkamal
    @amyrkamal 5 месяцев назад

    thank you professor

  • @alibinnaseer
    @alibinnaseer 10 месяцев назад

    Professor, we are dividing by h which approaches zero and we are also ignoring the terms that have h in them. Pls explain

    • @lucyla9947
      @lucyla9947 9 месяцев назад

      You can manipulate the equation to remove the h in the denominator.

    • @lucyla9947
      @lucyla9947 9 месяцев назад

      After plugging in f(a+h) and f(a) you can essentially simplify and remove the h from the derivative.

  • @SuperYoonHo
    @SuperYoonHo Год назад +1

    Nice Prof!

  • @aselim20.
    @aselim20. 2 года назад +1

    I wrote it.

  • @science-y9209
    @science-y9209 3 года назад

    1:50.. I didn't understand the graph.. for me the position WAS changing
    ... I didn't understand how the position didn't change but the speed did

  • @prithabasuneogi2325
    @prithabasuneogi2325 3 года назад

    You are a combination of a great scholar and a great teacher who is a great person to me

  • @BobDiaz123
    @BobDiaz123 5 лет назад +2

    45 years ago I took Calculus and I remembered the power rule, but I couldn't remember why the rule worked. Your explanation was wonderful.

  • @kikwito9916
    @kikwito9916 4 месяца назад

    He looks like Sheldon.

  • @SurinderKumar-os5il
    @SurinderKumar-os5il Год назад

    Sir
    Excellent regarding aaproach of secant. Limit. Tangent over lap
    Plz. Deffentiate tan( α)

  • @shinchan2778
    @shinchan2778 4 года назад

    👌👌👌👌👍very nice explanation sir!!!!!

  • @mathstaylor8786
    @mathstaylor8786 5 лет назад

    im useing khan academy and I am going of it because its a bit jumpy because the diffrentation and integration is in there ap calc sections and vice versa and I am not sure wethere to do the ap calc sections first or the diffrentation sections first?

    • @ProfessorDaveExplains
      @ProfessorDaveExplains  5 лет назад +2

      Well any calculus course will cover differentiation first, and then integration. But everything is laid it nicely and in the ideal order in my calculus playlist, so just head over there!

  • @gangabharatha
    @gangabharatha 4 года назад

    fine I an understand clearly. thank you.

  • @jashansehgal7347
    @jashansehgal7347 5 лет назад +1

    Best explanation

  • @Vinodbhalekar02
    @Vinodbhalekar02 Год назад

    You are simply Great 👍👍👍

  • @10OzGlove
    @10OzGlove 5 лет назад

    If only RUclips could provide a link to the next video which is an essential feature of this kind of platform, but apparently they're too busy "improving" their recommendation algorithm which acts as a beater for their sponsors .

    • @ProfessorDaveExplains
      @ProfessorDaveExplains  5 лет назад +1

      That's what the end screens are for. Watch to the end and it links to both the previous and next videos in the playlist.

    • @10OzGlove
      @10OzGlove 5 лет назад +1

      The speed of your reply :) Thank you. Turns out I had an extension to remove end cards because of certain youtubers using them inappropriately. It has been years that I don't see them any more so I had completely forgotten about them. Thanks again.

  • @abhinavdayal9512
    @abhinavdayal9512 4 года назад

    I played the intro infinite times without any limits :p

  • @avjeet888
    @avjeet888 6 лет назад +1

    you are a great teacher

  • @trieuhuyvan5020
    @trieuhuyvan5020 2 года назад

    😘😘😘😘😘😘😘😘😘😘😘😘😘😘😘😘

  • @yudhisthir7247
    @yudhisthir7247 5 лет назад

    Hello Dave, Thank you for providing great animated videos for helping the students.You are a great teacher. Now just to clear the idea,I have a query on this explanation of power rule. I know the power rule,which you have derived but how could you generalize only showing the pattern it follows , there should be some actual proof of the same. Just to back my question if you take the example , Natural numbers are not closed under subtraction but I will take example like this 9-8=1, 10-7=3 where these subtractions give me natural number , So can I conclude natural numbers are closed under subtraction ?. I am sure not. Then my generalizations fails here. Could you please help me to clear this.

    • @9308323
      @9308323 4 года назад

      I know this is late, but for every f(x) = x^n, x^n would *always* be cancelled out by -x^n. The rest of the function would *always* contain an h variable, which you can simply divide by the divisor. You would be left with every term having h in it with the exception of the first. Taking the limit, all the terms would equal zero with the exception of nx^(n-1), which is the derivative.

    • @chavezharding7820
      @chavezharding7820 2 года назад

      You could use the binomial theorem if you want to prove the rule rigorously.

  • @fleurblanche3727
    @fleurblanche3727 11 месяцев назад

    Thank you ❤

  • @sahalmohamedbare60
    @sahalmohamedbare60 10 месяцев назад

    Thank you❤❤❤❤❤❤

  • @simpdestroyer5743
    @simpdestroyer5743 23 дня назад

    Damn.

  • @ivan_ivan0v_
    @ivan_ivan0v_ Год назад

    Well explained! Thank you!

  • @liya2280
    @liya2280 Год назад

    This video helped me so much ❤️

  • @prachinainawa3055
    @prachinainawa3055 2 года назад

    Love your teachings 🥳

  • @matthewphares4588
    @matthewphares4588 Год назад

    This is excellent work.

  • @maxoh7399
    @maxoh7399 4 года назад

    Trigonometry please

  • @shahulsayyed9052
    @shahulsayyed9052 4 года назад

    How the derivative can be the slope of function at a particular point as slope refers to how steep something is and steepness is not a property of a point rather it is a property of a line?

    • @ProfessorDaveExplains
      @ProfessorDaveExplains  4 года назад

      there is a rate of change that can be determined for every point on a curve

    • @shahulsayyed9052
      @shahulsayyed9052 4 года назад

      But how can rate of change exist for a single point when rate of change fundamentally requires two points to be computed?

    • @ProfessorDaveExplains
      @ProfessorDaveExplains  4 года назад

      because it's not a single point, it's part of a curve

  • @mohamedmouh3949
    @mohamedmouh3949 4 года назад

    waw arab traduction. i love you bro❤️

    • @icequeen8820
      @icequeen8820 3 года назад

      ممكن اسم الموضوع بالعربي ؟

  • @지구과학천문학
    @지구과학천문학 4 года назад

    im just curious
    can anyone tell me what grade we learn these at school?

    • @ProfessorDaveExplains
      @ProfessorDaveExplains  4 года назад

      most people don't take calculus until senior year of high school, some don't take it at all.

    • @지구과학천문학
      @지구과학천문학 4 года назад

      @@ProfessorDaveExplains then ill be the one taking it!! thabks alot professor dave!

  • @Garentei
    @Garentei 2 года назад

    Why would I want to know the slope of the tangent line at s particular point?

    • @ProfessorDaveExplains
      @ProfessorDaveExplains  2 года назад

      Because that represents an instantaneous rate.

    • @Garentei
      @Garentei 2 года назад +1

      @@ProfessorDaveExplains And why would I want to know the instantaneous rate? (Good video btw)

    • @ProfessorDaveExplains
      @ProfessorDaveExplains  2 года назад +1

      Oh all kinds of reasons, analyzing the motion of objects, etc. Physics.

  • @MrMarkgyuro
    @MrMarkgyuro 4 года назад

    thank you! really understandable!

  • @aarututor
    @aarututor 4 года назад

    Awesome. May God bless you

  • @bhuvanasriramoji3835
    @bhuvanasriramoji3835 5 лет назад

    Thanks a lot🥳🤗🤗🤗🥳🥳💕

  • @shinystick241
    @shinystick241 3 года назад

    Thanks a lot!

  • @suziwallace5360
    @suziwallace5360 5 лет назад

    I love these videos

  • @nandakumarrajamanickam7812
    @nandakumarrajamanickam7812 5 лет назад

    fantabulous.

  • @sanelprtenjaca9776
    @sanelprtenjaca9776 5 лет назад

    I learn calculus, although it is curriculum for 4th grade. Thanks for perfectly explanation of the basics

    • @ProfessorDaveExplains
      @ProfessorDaveExplains  5 лет назад +3

      Calculus is taught in the last year of high school or in early undergraduate curriculum. No one learns this in 4th grade.

    • @sanelprtenjaca9776
      @sanelprtenjaca9776 5 лет назад

      @@ProfessorDaveExplains In Croatia (better schools) learn derivations, function compositons, limeses, etc. I don't know how deeply, but it is in the curriculum.
      And, watching your calculus videos, it looks quite simply.

    • @ProfessorDaveExplains
      @ProfessorDaveExplains  5 лет назад +1

      I don't believe you. Croatian elementary school students do not learn calculus. Feel free to email me syllabi or screenshots or images or any evidence to the contrary.

    • @sanelprtenjaca9776
      @sanelprtenjaca9776 5 лет назад

      @@ProfessorDaveExplains here in Croatia, (better) high schools have 4 grades 🤣😂
      So, I thought on 12th grade (8 + 4)

    • @ProfessorDaveExplains
      @ProfessorDaveExplains  5 лет назад +3

      Ah! Well that makes much more sense. Yes, I learned calculus in 12th grade, which is rather common. Classic moment of something being lost in translation it seems.

  • @omarel-nemr6506
    @omarel-nemr6506 5 лет назад

    great job

  • @adityashankar5267
    @adityashankar5267 5 лет назад

    What is f(x^-1)

    • @ProfessorDaveExplains
      @ProfessorDaveExplains  5 лет назад

      depends what the function is. you may want to go back to my algebra tutorials.

  • @vikasgupta8948
    @vikasgupta8948 6 лет назад

    Professor can you make a video on series?

  • @kavishkakavishka8117
    @kavishkakavishka8117 2 года назад +1

    Wowww! Professor dave has become my new best maths teacher of all ❤

  • @marekjanik9962
    @marekjanik9962 5 лет назад

    Most important part @5:02

    • @ronin6158
      @ronin6158 5 лет назад

      for *doing* calculus perhaps. For *understanding* calc its 2:17