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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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
Absolutely ! right. 🤞🏻
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.
I'm also a software application design architect, well it's fun to watch this video to go through the logic of human mind
Schools are stupid
I just cried of joy, how easy you made my life, Love You Professor Dave
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.
Very helpful videos, Prof. Dave!
How did you get 0 from f(x+h)-f(x) when x =5? Shouldn't it be 5+h-5/h?
but if f(x) = 5, then no matter what h is, the function is always equal to 5.
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.
woohoo! thanks in advance for the contribution :)
Alright bro, I'm a Mega Professor Pal.
awesome!!! you are a true hero!
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.
I noticed this too, just a little second to acknowledge that the audience is absorbing something. instinct of a great educator
Just As always , the best courses in the world
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.
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
👍👍👍👌👌
Exactly
what rank did you get in jee you did not qualify am i right
@@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.
Your explanation is of god level this is definitely the best calculus tutorial in the world I am very lucky to get it
this professor just took the entire 1 week of calculus in just 10minutes video... Godbless you sir for making these videos
Atheism bro.
PROFESSOR DAVE THIS VIDEO LITERALLY CHANGED THE WAY I VIEW CALCULUS THANK U SO MUCH 😭😭😭 seeing the difference quotient being visualized changed everything!!!!!
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.
Math is neat.
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.
I feel extremely lucky lol. My teacher explained it pretty much exactly like this. Seems that isn't the case everywhere sadly.
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.
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!
Teachers these days should learn from Prof. Dave. I've never seen a clearer explanation of derivatives.
Finally understood calculus.
THX❤
Thanks bunch! This really helped me a lot
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
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
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
No person can beat him or even compete with him
Professor Dave can make me watch a lesson about calculus even after I've passed
How does someone make calculus entertaining?!? Thank you for your videos
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.
Great vid, learned lots.
I swear everyone has a patreon in a post-adpocolyptic world.
An excellent explanation of how the power rule is derived.
> derived
I see what you did there
wonderful lectures. Thank you very much.
This is such a great explanation. Thank you so much!👏👏👏
my first time watching your maths videos and i was geared to work out comprehension question sine am used to watching your physics videos
some have it, some don't, depends on the nature of the lesson.
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
Thank you, math/chemistry/physics/biology jesus!
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professor please check your mail, i am seeking answer for that qus
3:42 Isn't h→a instead of h →0 in the formula for the tangent line?
Of course the distance become O
i came here for derivation of power rule and you just taught things that i already knew
Thank you sir for your dedication and for making this free!
Here come the details and expected actions for your refund
he is not explaining maths. he is developing maths nerds 😂
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.
5:23 why is h in denominator not 0 as h approaches 0
and we'll get 0/0?
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.
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
This video really helped me understand a crucial bit of calculus.
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.
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.
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
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?
You are such a good teacher! Thank you!!
God bless you man. Sincerely
7.59 where did the 3x go? i would appreciate if someone answers because I don't have time :(
Its elementary maths
Value of h is zero so 3x times h is also zero
Do you have any recommended worksheet or question bank? Please consider
me learning calculus a year earlier than i was going to so i can understand quantum mechanics: stonks
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.
I'm just 14 yrs old but I now understand calculus (differentiation), thanks to this channel.👍👍👍
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.
am just here for the intro
thank you professor
Professor, we are dividing by h which approaches zero and we are also ignoring the terms that have h in them. Pls explain
You can manipulate the equation to remove the h in the denominator.
After plugging in f(a+h) and f(a) you can essentially simplify and remove the h from the derivative.
Nice Prof!
I wrote it.
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
The green dot is the position
You are a combination of a great scholar and a great teacher who is a great person to me
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.
He looks like Sheldon.
Sir
Excellent regarding aaproach of secant. Limit. Tangent over lap
Plz. Deffentiate tan( α)
👌👌👌👌👍very nice explanation sir!!!!!
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?
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!
fine I an understand clearly. thank you.
Best explanation
You are simply Great 👍👍👍
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 .
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.
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.
I played the intro infinite times without any limits :p
you are a great teacher
😘😘😘😘😘😘😘😘😘😘😘😘😘😘😘😘
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.
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.
You could use the binomial theorem if you want to prove the rule rigorously.
Thank you ❤
Thank you❤❤❤❤❤❤
Damn.
Well explained! Thank you!
This video helped me so much ❤️
Love your teachings 🥳
This is excellent work.
Trigonometry please
check my trigonometry playlist
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?
there is a rate of change that can be determined for every point on a curve
But how can rate of change exist for a single point when rate of change fundamentally requires two points to be computed?
because it's not a single point, it's part of a curve
waw arab traduction. i love you bro❤️
ممكن اسم الموضوع بالعربي ؟
im just curious
can anyone tell me what grade we learn these at school?
most people don't take calculus until senior year of high school, some don't take it at all.
@@ProfessorDaveExplains then ill be the one taking it!! thabks alot professor dave!
Why would I want to know the slope of the tangent line at s particular point?
Because that represents an instantaneous rate.
@@ProfessorDaveExplains And why would I want to know the instantaneous rate? (Good video btw)
Oh all kinds of reasons, analyzing the motion of objects, etc. Physics.
thank you! really understandable!
Awesome. May God bless you
Thanks a lot🥳🤗🤗🤗🥳🥳💕
Thanks a lot!
I love these videos
fantabulous.
I learn calculus, although it is curriculum for 4th grade. Thanks for perfectly explanation of the basics
Calculus is taught in the last year of high school or in early undergraduate curriculum. No one learns this in 4th grade.
@@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.
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.
@@ProfessorDaveExplains here in Croatia, (better) high schools have 4 grades 🤣😂
So, I thought on 12th grade (8 + 4)
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.
great job
What is f(x^-1)
depends what the function is. you may want to go back to my algebra tutorials.
Professor can you make a video on series?
i did that! look earlier in the math playlist. in the 80s somewhere.
Professor Dave Explains yeah i get it
Wowww! Professor dave has become my new best maths teacher of all ❤
Most important part @5:02
for *doing* calculus perhaps. For *understanding* calc its 2:17