Lec 4 | MIT 18.01 Single Variable Calculus, Fall 2007
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- เผยแพร่เมื่อ 25 มิ.ย. 2024
- Chain rule
Higher derivatives
View the complete course at: ocw.mit.edu/18-01F06
License: Creative Commons BY-NC-SA
More information at ocw.mit.edu/terms
More courses at ocw.mit.edu
I appreciate how he thoroughly erases the blackboard.
you probably dont give a damn but does someone know a tool to log back into an instagram account?
I stupidly lost my login password. I would love any assistance you can offer me!
@Ernesto Matthias instablaster ;)
Yes it hurts me internally when prof. Jerison doesnt
I don't understand how the student in the audience thought that the u and v the professor drew looked similar.
I love this guy, he actually shows the FULL way to do things. And his proof for the product rule is beutiful
Absolutely true, his proof of the product rule was great!
Hii reply after 12 years
THANK YOU SO MUCH MIT, I REALLY APPRECIATE THIS...
This is great. I never got to learn this stuff in college so it is great to actually learn this at home in my own time which is video after video after video... Accelerated learning is great.
What I have learned from this section:
--- The product rule of derivatives and how to derive it
--- The quotient rule of derivatives and how to derive it
--- The chain rule (composition rule) of derivatives and how to derive it
--- The concept of higher derivatives and their notation
*applause* That was great.
I remember those rules as
"First d-second plus second d-first"
"Low d-high minus high d-low, all over low-squared"
"d-inside times d-outside."
I like how this free class even excludes the errors, making it ten times better than the paid for class above...those two mistakes with the x (should have been u) could greatly confuse one
Hii
This has got to be one of the hardest lectures to follow - even the students are baffled. "Where did that x^(-2n) come from?", "oh, well I just decided to perform 3 extra steps in my head without telling you what I was doing."
Smart fellow, but man, he's jumping all over the place without a lick of explanation.
I kinda agree that these lectures are harder to follow than the main professor’s, but I think that says more about the him than this substitute. The main professor is extremely good at explaining Calc one, and we are spoiled by that great lecturing. The x^(-2n) was a step he had explained in a previous context, and it’s only a simple application of exponent laws. It is not unreasonable to assume one could follow that line of reasoning without explicit explanation.
Also I just realized this comment is 8 months old…
@@copywrite9396 I think you nailed it on the head. This guy is probably on the "average calc teacher" level.
Prof. Jerison is a brilliant teacher. I've gone through this course like three times I think. I've gone through The Great Courses calc courses, and a few other teaching company courses. I've walked through the Khan Academy courses ... and I've read several calc books.
Prof. Jerison is, by far, one of the best teachers out there. There's just absolutely no question about that.
So, unfortunately, the stark contrast with this substitute is probably not all too fair to the substitute.
Thanks to MIT for sharing the knowledge for free!
Thank You MIT really appreciate what you doing.
Thanks for the videos mit! I watch em everyday...
I've just noticed that my teacher never taught me the entire steps...just the short cuts. This is great!
I am not in MIT, lol. I'm a second year biochem student in the UK teaching myself maths for fun. I am trying to watch all the lectures for this, and understand them. The problems you can find online however are a bit beyond me!
IDidactI I have a similar situation too, I go to med school but math and music are a hobby of mine.
Thank you MIT, very cool!
I love you so much MIT!!!
The v's have points on the bottom. The u's have little round things
Thanks for this MIT - a great aid for me.
I was skeptical seeing all the comments how he'd be. Considering how hard it is so jump en media res into a set of lectures and be clear, he does ok... But the moment he takes out the chain and they applaud him is one I'll never forget till I die. And that's why he gets the gold star. He may not have had the best technique in this one day, but I'll forever have that memory tethered into this topic.
I find him pretty good
Thanks for uploading this great video, MIT great professers
MIT is the best. thank you MIT !!!
Flawless victory!
The way he explained (ridiculed) different notations for higher order derivatives is priceless.
Please, can someone reach out to this professor and ask him to say something about different forms of partial derivatives.
Thanks again to the people who put this up. I liked this guy even better though the regular guy is good.
Thank you MIT. #Grateful.
He just simplified the algebra of it a little. He could have wrote u(X + deltaX)/v(X + deltaX) instead, more like the previous deduction of the product rule. However recall that u(X + deltaX) - u(X) = deltaU, and since we are missing the -u(X) term, u(X + deltaX) = deltaU(X) PLUS u(x). Hence as written, u + deltaU.
@Julian Hernandez I am indeed alive, alive and jealous of my former self that was self studying calculus on a summer break 😅 good luck with your studies!
AGGH Brilliant. My Calc teacher forgot the last part. Now I can make the whole thing rhyme!
I thought they murdered the other prof for not remembering the sin sum formula :D
7:19 to 8:09 were the most hopeless and confusing seconds of my life.
Yeah, same I had to resort to khan academy for a better proof. I quote, “Have I made a mistake here?”
@@ezrabaker3953 it's just because of a wrong bracket !!
I Really Like The Video From Your Chain rule Higher derivatives
The proofs of the values of (uv)' and (u/v)' are excellent : consider the variation of the product or the quotient, and then divide by the variation of x, and, at last, consider the limit wjen Dx goes to zero, which is the derivative.
Props to Saurav Bastola, th-cam.com/channels/7hkeE-LRyzietNCecWnHLg.html for the listed topics.
Lecture 1: Rate of Change
Lecture 2: Limits
Lecture 3: Derivatives
Lecture 4: Chain Rule
Lecture 5: Implicit Differentiation
Lecture 6: Exponential and Log
Lecture 7: Exam 1 Review
Lecture 9: Linear and Quadratic Approximations
Lecture 10: Curve Sketching
Lecture 11: Max-min
Lecture 12: Related Rates
Lecture 13: Newton's Method
Lecture 14: Mean Value Theorem
Lecture 15: Antiderivative
Lecture 16: Differential Equations
Lecture 18: Definite Integrals
Lecture 19: First Fundamental Theorem
Lecture 20: Second Fundamental Theorem
Lecture 21: Applications to Logarithms
Lecture 22: Volumes
Lecture 23: Work, Probability
Lecture 24: Numerical Integration
Lecture 25: Exam 3 Review
Lecture 27: Trig Integrals
Lecture 28: Inverse Substitution
Lecture 29: Partial Fractions
Lecture 30: Integration by Parts
Lecture 31: Parametric Equations
Lecture 32: Polar Coordinates
Lecture 33: Exam 4 Review
Lecture 35: Indeterminate Forms
Lecture 36: Improper Integrals
Lecture 37: Infinite Series
Lecture 38: Taylor's Series
Lecture 39: Final Review
@3:57: "And then I take "u" the way it is" That is just so beautiful, everyone should just be taken as they are
BEAUTIFUL 板书 ，neat！
hey um..., for some reason i cant watch any of the video in the web site you posted here. Whenever i click on the video icon it just loads but never get there.
This guy's great : ) thanks mit
Shout out to MIT for using subs instead of canceling class
Interesring the Math on MIT. Greattings from México
@prantare Because delta v represents the change of the value of v from the original v(x) value (in other words, delta v = "new" v - "old" v). As delta x goes to zero, "new" v and "old" v become the same (if v is continuous at this "old" v value, by the definition of continuity), and therefore the delta v approaches zero. Make sense?
Thank you MIT
You have a substitute 🙏🏼 I love it !!’
Did i miss anything,or did he actually prove quotient rule?
If anyone is still confused and wants to understand fully how to derive the product rule, check this lecture note from MIT OCW: ocw(dot)mit(dot)edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010/1.-differentiation/part-a-definition-and-basic-rules/session-9-product-rule/MIT18_01SCF10_Ses9c.pdf
+Queen ofCool Thanks. Khan Academy also gives an explanation I found good too www.khanacademy.org/math/differential-calculus/taking-derivatives/product_rule/v/product-rule-proof
+CharethCutestory Thank you. I will check it out.
Queen ofCool I'm happy to admit I was totally lost by his explanation lol. I knew all this stuff at one stage but have sadly forgotten. They're good lectures but this guys moves quickly and glossed over a lot of the algebra I thought. i feel a bit better as students in the class ask questions I would ask too.
He brought the actual chain!
At 44:00, can someone please explain to me where does he get the 2 from?. I understand that the x^1 comes from x^{n-n+1}, but the 2 that he adds?
Roberto Paredes here u=1,v=x^n so,v^2=(x^n)^2=x^2n ok?but here 1/v^2=x^-2n.that's it.
yah, basically sequences and also n! if you know what i mean.
X^n has n, n-1,n-2,...2
@zamaxx
MIT has one. 18.01 is the physics 1 videos. I believe there is also 18.02 and 18.03
Thank you so much.
I actually enjoyed the lectures by professor Miler much more than the lectures by professor Jerison.
very interesting lecture
at 16:28 why did delta v go to 0 with delta x ?
Excellent
thank you mit, supposing i am going to donate , how can i do this from iran?
great class =)
Never think in public....
Sehr gut professor
Best chalk I have ever seen
@29.30 the clapping was classic first time i heard applause in a calculus class
I think the derivative of U(x)V(x) ,we should add and substract u(x+deltax).v(x) instead of u(x) .....
It would be great if he could use less similar looking letters. Uvuvuvuvuvuvuvuvuvuvuvuvuvuvuvuvvuvuuvuvuvuvuvuvuvuvuvuvuvuuvuvuvuvuvuvuvuuvuvuvuvuvuvuvuuvuvuvuvuvuvuvuvuvuvuvuvuvuvuvvuvvuvuvu
The real question is why W is called "Double-u" when it is really a "double-v".
its just standard notation for chain rule
Natethesandman1 ıts quite common for mathematicians to use similar letters for similar things. Because sometimes you have so many letters that you have to memorize and it is hard to remember what is what. So we use similar letters for similar things to make it easier.
@@viktorzzxzIn french, W is actually called "double v"
The fun fact is the tutor said "they look different" and then messed them up a minute later
Thanks MIT
28:55 LOL momlevel explanation
Dear IDidactl,
Thank you very much for your reply. I understood now. Thanks again.
Are you taking his class as well?
Hi 10 years old comment
The problem here is that he's using "delta" which they've already been thought to call the "average" in place of the "d" which they know as "instantaneous" rate of change.
好正！！！
will read the notes to see what he left out on the proofs
thanks alot
hes "the substitute", but i sure liked how he explained that chain rule! very helpful.
Hey, so will ((d^a)/(dx)^a) * x^n be equal to (n!/(n-a)!) * n^(x-a)
The best thing about this class is you can take it Pass/Fail at MIT😊😊
All students should start out with a course in Group Theory
(& a *rigorous* treatment of Limits)
BEFORE taking Calculus.
Wait...Dr Miller's specialisation is in Algebraic Topology?!
How the f#@% did this lecture turn into such a confusing pile of computational BS, then?!
He should know better!!!
Well sometimes it is good to encapsulate things before exposing the students to it. The course is for all MIT students and assumes no other prereq, so it is normal not to treat limit formally (as in analysis) before teaching calculus.
Excelent!! :P
Low D-High Minus High D-Low, all over Low Squared we go!!
I never in my entire life had a substitute teacher teach a lesson. Professors just cancelled class, and in K-12, substitutes were pure babysitters.
can somebody explain me the explanation in 25:49 I can understand what he says but I'm confused about the text.
The first ratio on right side delta y/delta x = dy/dx as delta x goes to 0, here prof said delta t goes to 0 without explaining that it make delta x goes to 0 as well .
my only complaint is that he should've explained the notation he used for quantition .Because It's not as simple as it looks deltau stands for u(x+delta×)-u(x) and with the u which stands for u(x) those cancel out so there is the main equation for the derivatives without the explanation for not so sceptical students this could really go wrong I'm just saying
31:19 shouldn't that read dy/dt = 10cos(10) because x=10t => x' =10?
Or have I missed something?
+666munkynutz >> You mixed up the steps
Lets call: x= 10t
dy/dt = [sin(x)]' [x]'
dy/dt = cos(x) . x' = cos (10t) . (10t)' = [cos (10t)] [10] = 10 cos(10t)
+Tony Pham ahh I did, thanks :)
@ 17:04, it should be d/dv (1/v). If v really is a function of x, as he wrote, then he is using the chain rule which he hasn't introduced yet...
I am 3 years late, but what he has written is correct too. The chain rule and the quotient rule in this case give the same thing
13:13
How is delta(u/v) =(u+deltau)/(v+deltav) - u/v at x+deltax?
Nvm
this guy is so cool
lol that loud applause startled him a bit
Cool video
After you derive the product rule, you can easily prove the quotient rule by considering the derivative of the product f(x) * ( 1 / f(x) )
correction f(x)*(1/g(x))
I never trust substitutes.
GreyFace That's not very nice
yes you can if they keep a chain in their bag
dam such fun memories loved calculous
I'm just here to for the sound of chalk striking chalkboard.................
Great professor as always at MIT. However, I wonder if there is a mistake here for the deduction that he did (time: 13:28/46.03Min)...I did not get it.
Delta (U/V)=(U+DeltaU)/(V+DeltaV)-U/V
46 minutes. woah. can i haz crash course?
But at sin(x)^10 couldn't we use the product rule? Like take the derivative of (sinx*sinx*sinx*sinx*sinx*sinx*sinx*sinx*sinx*sinx) and use it as d/dx(abcdefghij)=a'bcdefghij+ab'cdefghij+etc etc where a, b, c, d, e, f, g, h and i are all equal to sinx?
You could do that and you will get 10((sinx) ^ 9)(cosx), but imagine if you had to take the derivative of sin(x) ^ 100 of sin(x) ^ 1000. Using the product rule with those functions would take forever! You are saving a lot of time computing the derivative using the chain rule.
i used the varibles f(x) and g(x) instead of u and v
16:19 . Why did he say that "v is continuos"
I need a help:
At 16:16 min, I still didn't get how Δv goes to zero
because v is a continous which means the change in v i.e delta v is so small which tend -------> to zero 0
what the hell is this cross multiplying stuff with the quotient rule proof can someone explain his way from 1st to 2nd line :(
He is just finding the common denominator v(v+^v) and dividing by (v+^v) which is v, then multiplying this answer by the first nominator, he then does the same with the second part, v(v+^v)/v = (v+^v) and multiplies this by the second nominator (v) which is just the same as cross multiplying.
I almost had a brain failure trying to figure out how it was cross multiplication lol. He should have said LCD. :) I'm loving these videos so far.
I didn't understand it until I saw the chains.
If s/t is velocity what does ds/dt mean?
instantaneous velocity
S/t is average velocity.
Respect for 9th graders watching this
thanks !
I am 8th grader
Liked that teacher! he is funny ^^
great class \o
Wish I went to MIT.
So does every one!
I think the regular professor is better! Well, I liked the way he taught the chain rule, but I didn't like the way he proved the product rule and the quocient rule.
you can just substitute new u and v in the formula for (uv)prime
just put 1/v instead of v and you get (u/v)prime
If I don’t remember anything about Calculus, it’s the Chain Rule 😀 Don’t ask me anything else. Well! We did do a lot of Matric Algebra 😅