Most math professors are not good at explaining, much less making any sense. When a professor, like professor Leonard is found, he is worth his weight in gold. Thank you for making difficult theories easy to understand.
I cannot imagine where I would be in my calc class without these videos. I had been stuck for weeks, while searching for an outside source to help me. And these videos are really saving me. Thank you so much not only for being an amazing professor, but taking the little extra effort to make it possible for me to "be in your classroom".
@@aemv12 I would not change Pr Leonard with any channels,but for other subjects that channel is so useful thank you for recommending it.I found the channel I really needed for physics and chemistry.Just wanted to say thank you!!!
For comment about cancellation of "du," here is what I hope you will find to be an adequate response: I suppose it does suffice for a Calc 1 student as the COMPLETE proof takes some time and as soon as epsilon is put on the white-board, eyes start to gloss over... BUT, in a simple proof using Leibniz notation (as this is), this actually DOES represent a fraction as we would think of it and DOES "Cancel-Out," but only BEFORE taking the limit! So, if you were to use the definition of dy/du*du/dx AS A LIMIT, the "du's" would cancel (this works only if Delta u and Delta x BOTH approach Zero, please read the last few sentences). The reason why I tell the students to think of it that way is because in the proof, it is Delta 'u' that "Cancels." It absolutely IS DIFFERENT FROM du, because "du" would contain with it the idea of a limit, but that is where if comes from. You would define Delta 'u' to be the change in g(x), as x changes, and Delta 'y' to be the change in f(u) as g(x), which is 'u', changes. Because we are finding the derivative of f(x), and not f(u), that 'u' acts as a dummy variable. Its is introduced in Leibniz notation as a fraction. If you were to do the definition of a derivative, the one with all the limits, you would see that the numerator would contain "change in f(x)" and the denominator would contain "change in x" One way to develop the Chain Rule (derivative of a composition) is actually to use fractions, as we know them, INTO that definition. Notation-wise, we get dy/du*du/dx (AFTER taking the limits). Now, the reason why I say "One way" and "Simple proof" is because that proof is not complete and only proves one case where Delta U approaches zero when Delta X approaches zero. To deal with the case where is DOESN'T, you need the more involved proof that more people are familiar with. But thank you very much for the comment and I hope this clears up any ambiguity. P. Leonard
You are great i follow you lecture on youtube. I am taking an online class and let me tell you Because of your lecture i have an A in the class. Thank you so much your wondeful!!! Also do you also teach trig? because if you do please post the videos because i am taking trig next semester and i would like to follow your video to get an A in trig too.. Thank you
Professor Leonard, I really want to thank you for posting these videos. They have been a life-saver for my online Statistics course. The videos through my school are atrocious, and you really bring the material to life and explain the concepts in a way that people can relate to and easily understand. You're an awesome teacher! By any chance do you have any videos for Statistics past Lecture 8.6, or do you offer tutoring?
Hi Professor, been working my way through your Calc 1 playlist. Just wanted to say a big thank you to all the time and effort you've so obviously put in to creating a clear, fluid and easy to understand course! I was curious is there anyway I could get my hands on the homework your students do in class? PDFs or anything would do! Thanks again, peace
+Professor Leonard first thank you. Quick question, Wolfram Alpha gives a different derivative solution for the example (x^3+csc(x^3))^1/2 @ 1:18:00 would be great if you could clarify this? Thank you
37:37 I love how keep reminding what calculus is about. I was once a student who don't lost track of what derivative actually is when I was in my freshmen days, that's why I've never understand calculus before. But now that I'm re-learning it, everything is starting to make sense because of how amazing professor Leonard's teaching skill is. I'm really grateful for him.
He somehow always manages to indirectly answer all of my would-be questions and that is--in my opinion--the hallmark of a great teacher. Keep on doing what you're doing, professor.
I'm new to Calculus, and I'm studying it by myself, at the age of 31. If only this had been my very first lesson on the Chain Rule, it would have spared me a few weeks of frustration and self-doubt. I get it now. This is such a great resource! Thank you so much!
I have a degree in Biochemistry and am reviewing what I learned. I wish Professor Leonard had taught me. I spent many days trying to figure out stuff that was badly taught. It doesn't matter? It does to me. Calculus was the most interesting part of my education. It was so poorly taught, it didn't seem like it mattered to them.
Professor leonard, I jusy wanted to thank you for posting these many videos for us all. I've been taking Calculus 1 online and to say it has been a massive headache (since the book likes to skip steps, thus causing me up to 2 hours on a single problem through research) is an understatement. The way you teach is very effective and I like that you explain everything and actually take the time to clarify if needed. A+ for teaching!! Keep up these awesome videos you have helped me a lot and forever will I be grateful.
I'll be going for my postgraduate in data science next year. i got an undergraduate degree in engineering but forgot all my math concepts after working in industry. i am slowly working my through these calculus lectures, and soon i will be referring to the stats lectures in this course. I am immensely thankful to you Professor for posting such easy to understand lectures online for free for our consumption. If i was not saving up as much as i can to minimize my loan, i would have donated. but the first thing i will do once i get my job after post-graduation will be to come back here and donate. once again. thank you so much for everything.
I wanted to comment and just thank Professor Leonard for teaching Calculus in an inviting manner. I have struggled in the past with understanding calculus, and his positive demeanor, encouragement and teaching skills have allowed me to believe in myself again.
My usual problem is that I don't catch on in one lesson. It's like watching a movie; I understand it better if I see it again from the beginning. But the teacher does one lesson and assumes we're all experts at the end and will only do tutorials on specific problems. So this is a perfect solution. I can pause if I need time to comprehend, rewind, and watch it as many times as I need to understand. This is great.
8 years after the video has been posted and I am here learning more than I do in my match class at the University. Thank you, prof, for making things easy.
I'm busy doing a "distance learning course" for calculus and my text book spends about 1 page worth of examples and explaining it. The worst part is all the examples are so similar that if you given something slightly different you don't know how to approach it. Thank you so much Professor Leonard, probably wont see this, but I really appreciate the work you put in to math and the care you show. Going at a nice pace, explaining it in depth and doing a multitude of varying examples. without you I would be having breakdowns not know what to do, with you I feel confident and always excited to do more and learn more math. much love from South Africa :)
I would be tempted, if I were a Maths Teacher, to rather ask any pupil that does not understand to raise their hand. This way I should be able to instantly identify those that struggle. Eg, "2x3+5=11. Is anyone unhappy about this ?" What I`m getting at is that I would want to know those people that don`t get it, in order to help them. Those that get it don`t need help. Naturally psychology may come into play here. Raising ones hand and admitting that one does not get it may not be an easy thing to do when one is surrounded by people that do get it. Being the odd one out among ones peers is never easy.
Been years since undergrad and I wanted to retake calc to bypass GMAT for grad school (and to get a better grade). I was about to withdraw because I absolutely didnt want to get less than a B in the course and things were looking pretty slim for the next exam so I was thinking about dropping to save the GPA and try again next term but this channel literally clarified 3 weeks of material and I only watched 4 videos (+ redid all the homework with this newfound clarity). I'm about to use the channel in tandem with course material for the next 2 exams. If it all pans out, acceptance and completion of the program, I'm not above inviting you to my graduation in a few years hahaha. Thanks, Prof Leonard. Wish all my teachers were like you. Masterclass in how to lecture.
I am so grateful for you and these videos! Thank you sooooooooooo much! My instructor zooms through the lecture and I get lost, but I watch your video and its ALL clear! Thank you! Please don't stop helping us students out(for free!)!
Leonard, you are literally a lifesaver. I am so glad I am able to look through your videos and understand calculus concepts better. Thank you so much for being able to explain things so eloquently. You are the best.
Hello Professor Leonard. My name is Luis. I took calculus back in high school and I scored so poorly on the college board exams. My teacher taught me well but I failed to understand some parts. My teacher did explain to me what these rules and theorem mean. He was just like you. He said the steps matter the answer doesn't. As I recall. I'm going to take Calculus 1 at a university. I'm taking notes and I am watching your videos 100% all the way to the end. Your videos have become a great help so far. Thank you for your videos. I know you aren't getting paid. I am also a math major. As an adult I understand this more versus when I was in high school. Thanks you. Unfortunately, you won't reply haha because you haven't looked back at your videos but I express my gratitude to you. Thank you!!!!!
I'm glad that you cover both methods of the Chain Rule. I have a crappy teacher who has only taught the chain rule through u-substitution, completing neglecting the now significantly easier derivative of f(g(x)). Even then, you explained u-subbing far better than she did. You did me a big service, and I'm sure I'm far from the only you did this for.
I won't physically be taking a calculus class for another year but I know that when I do take one, Professor Leonard would've been my first calc teacher. Thank you so much.
I got 100s on all my Calc 1 homeworks thus far this semester, and as soon as the test was put in front of me I blanked... I bombed it really hard despite all my studying... this is helping so much with helping me to *really* understand it. I can't thank you enough for uploading all of this! It's funny because despite my rough relationship with math throughout the years, I've never hated it -- I've always had a very healthy appreciation for it. The greatest things people have ever done were done because of math. I've always thought math is one of the coolest things ever; it's both natural and artificial. Thanks again for this series!
Great lecture. Nice and slow so I can get it, and every detail is explained. This really helps me retain the knowledge when I know the how's and why's.
So like others, I am taking an online class, but I just want to thank you as well... you're amazing. "follow the dy/dx". Makes SO much more sense. Thanks you so much for making these videos.
Introduction and Relevance of the Chain Rule ● [0:00]. Introduction to the Chain Rule and its importance in calculus. ○ The Chain Rule as the last main differentiation rule. ○ Applications of the Chain Rule and its use in real-world problems. ○ The three fundamental differentiation rules: Product Rule, Quotient Rule, and Chain Rule. ○ The Chain Rule as a method for deriving compositions of functions. Motivation and Examples of the Chain Rule ● [1:19]. Review of the concept of function compositions. ● [1:26]. Example illustrating the need for the Chain Rule: derivative of (3x² - 4)^100. ○ Difficulty of deriving the expression (3x² - 4)^100 using traditional methods. ● [2:31]. The Chain Rule as an efficient solution for this type of derivative. Explanation of the Chain Rule Through an Example ● [3:06]. Expression of (3x² - 4)^100 as a composition of functions. ● [3:40]. Method to identify the inner and outer functions in a composition. ● [4:01]. Definition of y = u^100 and u = 3x² - 4. ● [4:47]. Verification that the composition of y and u results in the original expression. ● [5:07]. Calculation of dy/du and du/dx. ○ Objective: find dy/dx. ● [7:50]. Justification of Leibniz notation for the derivative. ○ *The Chain Rule*: dy/dx = (dy/du) ⋅ (du/dx). ● [8:45]. Application of the Chain Rule to the example (3x² - 4)^100. Interpretation and Generalization of the Chain Rule ● [12:06]. Simplification of the differentiation process using the Chain Rule. ● [13:20]. Definition of the Generalized *Power Rule*. ● [14:05]. Demonstration of the Chain Rule from the definition of compositions. d/dx [(f(x))^n] = y = u^n, u = f(x) dy/du = n ⋅ u^(n-1); du/dx = d/dx [f(x)] dy/dx = (dy/du) ⋅ (du/dx) dy/dx = n ⋅ u^(n-1) ⋅ d/dx [f(x)] = n ⋅ {f(x)^(n-1)} ⋅ d/dx [f(x)] ● [16:26]. *General formula for the derivative of a function raised to a power*: ○ d/dx [(f(x))^n] = n ⋅ {f(x)^(n-1)} ⋅ d/dx [f(x)] ● [18:40]. Statement in simple language of the Generalized Power Rule. 1. Multiply by the exponent n. 2. Keep the base function raised to n−1 (i.e., reduce the exponent by one). 3. Multiply by the derivative of the base function. Examples and Applications of the Chain Rule ● [19:50]. y = (x³ - 2x + 38)⁴ find y' ○ Importance of recognizing function compositions when applying the Chain Rule. ○ Common errors when applying the Generalized Power Rule (forgetting to subtract 1 from the exponent). ○ Use of parentheses to indicate the multiplication of the inner derivative. ○ Relationship between the Generalized Power Rule and the simple Power Rule. Additional Examples: Combination of Differentiation Rules ● [24:25]. d/dx [(x)⁶] ○ Could ypu do the general power rule? ● [26:39]. d/dx [(2x - 3)(x² - 5)³] ○ Example combining the Product Rule and the Generalized Power Rule. ○ Determining which rule to apply first: the Product Rule applies to the entire expression. ○ [29:38]. Application of the Product Rule as the main rule ● [27:28]. d/dx [(x² - 1)² / (x + 4)] ○ Identifying the necessary differentiation rules: Chain Rule (or Generalized Power Rule) and Quotient Rule. ○ Order of application: Quotient Rule first ●[28:00]. d/dx [(x² - 1)² / (x + 4)]⁵ ○ Order of application: Generalized Power Rule, then Quotient Rule, and finally another Generalized Power Rule within the Quotient Rule. ●[29:38]. d/dx [(2x - 3)(x² - 5)³] ○ Application of the Product Rule as the main rule ●[39:05]. d/dx [ √(5x² - 1)] ○ Application of the Chain Rule to derivatives involving square roots. ○ Representing square roots as fractional exponents to facilitate differentiation. Extension of the Chain Rule to Trigonometric Functions ●[43:08]. Derivative of the function cos(x⁴) using the Chain Rule. ○ Identifying the outer function (cosine) and the inner function (x⁴). [27] ○ Applying the Chain Rule: derive the outer function, keep the inner function unchanged, and multiply by the derivative of the inner function. ●[46:07]. *Statement of the Chain Rule for trigonometric functions*: ○ d/dx [f(g(x))] = f'(g(x)) ⋅ g'(x). Additional Examples and Practice with the Chain Rule ● [48:22]. Practice with examples combining the Chain Rule, Product Rule, and Quotient Rule. ● [49:09]. Example d/dx [sin(4x⁵)] ○ Example of a trigonometric function with a composite argument ○ Identifying that this is a Chain Rule. ○ Recognizing the composition of functions: the outer function is sine and the inner function is 4x⁵. ○ Applying the Chain Rule: derive the outer function, keep the inner function unchanged, and multiply by the derivative of the inner function. ● [52:15]. Example d/dx [cos²(x⁴)] ○ *Recommendation*: analyze the structure of a problem before applying differentiation rules. ○ Distinction between the general Chain Rule and the Generalized Power Rule. ○ In this context, the Generalized Power Rule refers to the Chain Rule applied to a function raised to a power. ● [1:01:08]. Example d/dx[tan(3x² - 2x)] ○ If d/dx[tan⁴(3x² - 2x)] then first general power rule ○ Identifying the Chain Rule: the outer function is tangent and the inner function is 3x² - 2x. ○ Applying the Chain Rule: derive the outer function, keep the inner function unchanged, and multiply by the derivative of the inner function. ○ Common errors when applying the Chain Rule to trigonometric functions: forgetting to multiply by the derivative of the argument. ● [1:06:37]. Example d/dx[(x³ + csc(x³))^(-1/2)] ○ Simplifying the expression by moving the negative exponent to the denominator. ○ Identifying the Generalized Power Rule: the outer function is raising to the power of -1/2 and the inner function is the expression within the parentheses. ○ Applying the Generalized Power Rule: bring down the exponent, keep the internal expression unchanged, and multiply by the derivative of the internal expression. ○ To derive the internal function, it is necessary to apply the Sum Rule, Product Rule, and Chain Rule. ● [1:19:40]. Example d/dx[(3 + x²⋅cot(x²))^(-3)] ○ Example illustrating the convenience of simplifying expressions before applying the Quotient Rule. ○ In this case, d/dx[1 / (3 + x²⋅cot(x²))^(3)], it is recommended to move the denominator to the numerator with a negative exponent to apply the Generalized Power Rule. ○ Avoid using the Quotient Rule, as it can complicate the differentiation process. ● [1:31:20]. Exercise d/dx [(1 + cos(x²)) / (1 + sin(x²))] ○ If d/dx [(1 + cos(x²)) / (1 + sin(x²))]⁵ then apply general power rule, quotient rule and chain rule ● [1:32:10]. Exercise d/dx [x²⋅sin(3x)] ○ Product rule and then chain rule
Awesome video. My book made it impossible to comprehend and my teacher didn't do a very good job explaining it. After watching the video I now understand what I need to know. Thanks professor!
1:14:01 I had a hard time seeing this. I had to take one more step in order to clarify it for myself that the derivative of x^3 is not going to be multiplied into the 3x^2. I did: d/dx[x^3] + d/dx[csc(x^3)] and broke those two things up first so I could see that one derivative didnt have anything to do with the other derivative. Im glad someone in class asked so that you could clarify that one. Very helpful videos. Thanks Prof. Leonard
I am so happy I found your videos, you don't skip steps and are very clear. Just wanted you to know that you are having a positive impact on my learning. Thanks a million from Canada.
Thank you so much Prof,am from Mombasa Island in Kenya-Africa...I really appreciate your efforts ,for sure have learned alot about the rules in differentiation.may God bless you abundantly
I am glad that I found such an amazing teacher in my life. Thanks for making an extra effort to let students like us get the chance to fill our bosoms with the shower of the knowledge that you give.
You honestly do not know how appreciated you are, you are a wonderful mathematician, you have cured me of so many intangible curiosities i had. You are intuitive, fun and an amazing inspiration to everyone, thank you SOOO much Professor Leonard, for making these videos, they legitimately mean a lot.
These videos are so easy to digest that I actually watched beyond what my calc professor has taught us so far, and I didn't even notice it. I checked my notes and my professor stopped at sum and difference derivatives and I'm already on the chain rule. I feel like that's truly the mark of a great teacher when you don't feel so overwhelmed you have to stop at one or two lessons.
I finally get chain rule.. I must have read the stupid text book a dozen times and watched the class lesson we are given at least three times.. Made zero sense as to what exactly I was doing.... Now I get it! Wow thanks man!
Never able to learn calculus in my college, but after my college i.e vacation after last semester of college learning calculus for no reason I mean I will sure not get marks for it now, but doing it cause it is fun cause of Leonard, and yes , today I am staying awake whole night watching his video.
Thank you so much prof.. I have looked many videos on you tube and definitely you are among the best if not the best. You've made it easy to understand.
Need to keep going gotta complete all calculus by 31 december I was taught all this in the beginning of the year (class 11 ) but not so clearly now I am getting everything clearly.. Thanks a lot
man id be screwed if I hadnt stumbled on these vids. The why college is explaining this leaves a lot to be desired. Thanks for saving my grade Professor Leonard!
Thank you for a very clear explanation of the chain Rule. I believe there was a small error at 34:15 where the final d(x^2)dx should be 2 rather than 2x. But it didn't distract from the main point of the discussion so thanks again for providing such a good series on calculus. Jerry E Shepherd
I have one question for the last problem where the quotient rule was used. Why did you switch from brackets to parenthesis? Couldn't you keep using brackets for the entire problem?
Hi Professor Leonard, just a heads up, but at 58:54 couldn't you use the sin double angle formula to further siplify the expression and end up with -4x^3(sin(2x^4))? Love your videos, and grinding through the playlists! Thank you so much!
![Blox Fruits Dragon Rework Update [Full Stream]](http://i.ytimg.com/vi/EqDAp8udhm0/mqdefault.jpg)
I have watched these videos so religiously and so much that I recognize this one student's laugh. I hope he is doing great wherever he is now...
yeah me too brother, several of their accents, laughs, tones... Hope all of them are doing their best
Most math professors are not good at explaining, much less making any sense. When a professor, like professor Leonard is found, he is worth his weight in gold. Thank you for making difficult theories easy to understand.
I cannot imagine where I would be in my calc class without these videos. I had been stuck for weeks, while searching for an outside source to help me. And these videos are really saving me. Thank you so much not only for being an amazing professor, but taking the little extra effort to make it possible for me to "be in your classroom".
go to Michel van Biezen. You'll save time.. and learn more
@@aemv12 wait why are you here then??
@@aemv12 I would not change Pr Leonard with any channels,but for other subjects that channel is so useful thank you for recommending it.I found the channel I really needed for physics and chemistry.Just wanted to say thank you!!!
@@nigarabdullayeva7902 Hey! I was wondering what the channel was that you used for physics and chemistry?
@@josealvarez9451 leonard susskinds stanford playlists for physics ruclips.net/p/PL6i60qoDQhQGaGbbg-4aSwXJvxOqO6o5e
For comment about cancellation of "du," here is what I hope you will find to be an adequate response:
I suppose it does suffice for a Calc 1 student as the COMPLETE proof takes some time and as soon as epsilon is put on the white-board, eyes start to gloss over... BUT, in a simple proof using Leibniz notation (as this is), this actually DOES represent a fraction as we would think of it and DOES "Cancel-Out," but only BEFORE taking the limit! So, if you were to use the definition of dy/du*du/dx AS A LIMIT, the "du's" would cancel (this works only if Delta u and Delta x BOTH approach Zero, please read the last few sentences). The reason why I tell the students to think of it that way is because in the proof, it is Delta 'u' that "Cancels." It absolutely IS DIFFERENT FROM du, because "du" would contain with it the idea of a limit, but that is where if comes from. You would define Delta 'u' to be the change in g(x), as x changes, and Delta 'y' to be the change in f(u) as g(x), which is 'u', changes. Because we are finding the derivative of f(x), and not f(u), that 'u' acts as a dummy variable. Its is introduced in Leibniz notation as a fraction. If you were to do the definition of a derivative, the one with all the limits, you would see that the numerator would contain "change in f(x)" and the denominator would contain "change in x" One way to develop the Chain Rule (derivative of a composition) is actually to use fractions, as we know them, INTO that definition. Notation-wise, we get dy/du*du/dx (AFTER taking the limits). Now, the reason why I say "One way" and "Simple proof" is because that proof is not complete and only proves one case where Delta U approaches zero when Delta X approaches zero. To deal with the case where is DOESN'T, you need the more involved proof that more people are familiar with. But thank you very much for the comment and I hope this clears up any ambiguity.
P. Leonard
You are great i follow you lecture on youtube. I am taking an online class and let me tell you Because of your lecture i have an A in the class. Thank you so much your wondeful!!!
Also do you also teach trig? because if you do please post the videos because i am taking trig next semester and i would like to follow your video to get an A in trig too..
Thank you
Professor Leonard, I really want to thank you for posting these videos. They have been a life-saver for my online Statistics course. The videos through my school are atrocious, and you really bring the material to life and explain the concepts in a way that people can relate to and easily understand. You're an awesome teacher! By any chance do you have any videos for Statistics past Lecture 8.6, or do you offer tutoring?
Hi Professor, been working my way through your Calc 1 playlist. Just wanted to say a big thank you to all the time and effort you've so obviously put in to creating a clear, fluid and easy to understand course! I was curious is there anyway I could get my hands on the homework your students do in class? PDFs or anything would do! Thanks again, peace
+Professor Leonard thank you sir
+Professor Leonard first thank you. Quick question, Wolfram Alpha gives a different derivative solution for the example (x^3+csc(x^3))^1/2 @ 1:18:00 would be great if you could clarify this? Thank you
Professor Leonard can teach someone with zero math knowledge to become a calculus god. Thank you sir!
37:37 I love how keep reminding what calculus is about. I was once a student who don't lost track of what derivative actually is when I was in my freshmen days, that's why I've never understand calculus before. But now that I'm re-learning it, everything is starting to make sense because of how amazing professor Leonard's teaching skill is. I'm really grateful for him.
He somehow always manages to indirectly answer all of my would-be questions and that is--in my opinion--the hallmark of a great teacher. Keep on doing what you're doing, professor.
I'm new to Calculus, and I'm studying it by myself, at the age of 31. If only this had been my very first lesson on the Chain Rule, it would have spared me a few weeks of frustration and self-doubt. I get it now. This is such a great resource! Thank you so much!
same boat here (32 years old). makes me so made that I am paying my university so much money!
31 here, same boat... pissed about my bad instruction, thankfully we have Professor Leonard!
i'm 36 and learning this for the first time as well
I have a degree in Biochemistry and am reviewing what I learned. I wish Professor Leonard had taught me. I spent many days trying to figure out stuff that was badly taught. It doesn't matter? It does to me. Calculus was the most interesting part of my education. It was so poorly taught, it didn't seem like it mattered to them.
13 here, same boat.
Professor leonard, I jusy wanted to thank you for posting these many videos for us all. I've been taking Calculus 1 online and to say it has been a massive headache (since the book likes to skip steps, thus causing me up to 2 hours on a single problem through research) is an understatement. The way you teach is very effective and I like that you explain everything and actually take the time to clarify if needed. A+ for teaching!! Keep up these awesome videos you have helped me a lot and forever will I be grateful.
I'll be going for my postgraduate in data science next year. i got an undergraduate degree in engineering but forgot all my math concepts after working in industry. i am slowly working my through these calculus lectures, and soon i will be referring to the stats lectures in this course. I am immensely thankful to you Professor for posting such easy to understand lectures online for free for our consumption. If i was not saving up as much as i can to minimize my loan, i would have donated. but the first thing i will do once i get my job after post-graduation will be to come back here and donate.
once again. thank you so much for everything.
I wanted to comment and just thank Professor Leonard for teaching Calculus in an inviting manner. I have struggled in the past with understanding calculus, and his positive demeanor, encouragement and teaching skills have allowed me to believe in myself again.
My usual problem is that I don't catch on in one lesson. It's like watching a movie; I understand it better if I see it again from the beginning. But the teacher does one lesson and assumes we're all experts at the end and will only do tutorials on specific problems. So this is a perfect solution. I can pause if I need time to comprehend, rewind, and watch it as many times as I need to understand. This is great.
Thank God for these videos! They're really saving me in my class. Thank you professor.
8 years after the video has been posted and I am here learning more than I do in my match class at the University.
Thank you, prof, for making things easy.
Bruh I’m doing this in 8th grade.
@@mortiloniperpomontinson bro what country are you from?
For the first time ever I was able to enjoy a math class and completely understand the material, thanks dude you’re best math teacher I’ve ever had
I'm busy doing a "distance learning course" for calculus and my text book spends about 1 page worth of examples and explaining it. The worst part is all the examples are so similar that if you given something slightly different you don't know how to approach it. Thank you so much Professor Leonard, probably wont see this, but I really appreciate the work you put in to math and the care you show. Going at a nice pace, explaining it in depth and doing a multitude of varying examples. without you I would be having breakdowns not know what to do, with you I feel confident and always excited to do more and learn more math. much love from South Africa :)
am i the only one that raise my hand too when he ask 'raise your hand if you're ok with this' lol
I always do...haha
HAHAHAHAH so do I
I feel the sensation creeping up and I realize I'm at home.
I would be tempted, if I were a Maths Teacher, to rather ask any pupil that does not understand to raise their hand. This way I should be able to instantly identify those that struggle.
Eg, "2x3+5=11. Is anyone unhappy about this ?"
What I`m getting at is that I would want to know those people that don`t get it, in order to help them. Those that get it don`t need help.
Naturally psychology may come into play here. Raising ones hand and admitting that one does not get it may not be an easy thing to do when one is surrounded by people that do get it. Being the odd one out among ones peers is never easy.
Who wants to say " I am dumb" ??
I wish I had you as a teacher for Calculus
once I have my six figure job, I'm contributing what I paid for math courses to your patreon. Thank you so much for your hard work.
Been years since undergrad and I wanted to retake calc to bypass GMAT for grad school (and to get a better grade). I was about to withdraw because I absolutely didnt want to get less than a B in the course and things were looking pretty slim for the next exam so I was thinking about dropping to save the GPA and try again next term but this channel literally clarified 3 weeks of material and I only watched 4 videos (+ redid all the homework with this newfound clarity). I'm about to use the channel in tandem with course material for the next 2 exams. If it all pans out, acceptance and completion of the program, I'm not above inviting you to my graduation in a few years hahaha. Thanks, Prof Leonard. Wish all my teachers were like you. Masterclass in how to lecture.
it's the first time I am excited to do calculus problems on my own, thanks Prof. Leonard
I am so grateful for you and these videos! Thank you sooooooooooo much! My instructor zooms through the lecture and I get lost, but I watch your video and its ALL clear! Thank you! Please don't stop helping us students out(for free!)!
Leonard, you are literally a lifesaver. I am so glad I am able to look through your videos and understand calculus concepts better. Thank you so much for being able to explain things so eloquently. You are the best.
Professor Leonard, i have understood Calculus better now than when I took Calculus in University. You are the BEST!
Thank you so much for these. Even 10 years later you're saving calc students! You make it so much easier to understand!
Hello Professor Leonard. My name is Luis. I took calculus back in high school and I scored so poorly on the college board exams. My teacher taught me well but I failed to understand some parts. My teacher did explain to me what these rules and theorem mean. He was just like you. He said the steps matter the answer doesn't. As I recall. I'm going to take Calculus 1 at a university. I'm taking notes and I am watching your videos 100% all the way to the end. Your videos have become a great help so far. Thank you for your videos. I know you aren't getting paid. I am also a math major. As an adult I understand this more versus when I was in high school. Thanks you. Unfortunately, you won't reply haha because you haven't looked back at your videos but I express my gratitude to you. Thank you!!!!!
You're very welcome! Best of luck in Calc 1 :)
I'm glad that you cover both methods of the Chain Rule. I have a crappy teacher who has only taught the chain rule through u-substitution, completing neglecting the now significantly easier derivative of f(g(x)). Even then, you explained u-subbing far better than she did. You did me a big service, and I'm sure I'm far from the only you did this for.
These video's saved my life. I did catch myself raising my hand like I was in the class! OMG.
Over my course of doing this Calculus playlist, I've found that Algebra, with all of its rules and properties, is the closest thing to magic.
I won't physically be taking a calculus class for another year but I know that when I do take one, Professor Leonard would've been my first calc teacher. Thank you so much.
how did i learn all this in one video yet my teacher couldn't teach me this in a year. Shows how talented and also buff professor Leonared is
I got 100s on all my Calc 1 homeworks thus far this semester, and as soon as the test was put in front of me I blanked... I bombed it really hard despite all my studying... this is helping so much with helping me to *really* understand it. I can't thank you enough for uploading all of this! It's funny because despite my rough relationship with math throughout the years, I've never hated it -- I've always had a very healthy appreciation for it. The greatest things people have ever done were done because of math. I've always thought math is one of the coolest things ever; it's both natural and artificial. Thanks again for this series!
I don't know how I would learn Calculus if it weren't for your wonderful, amazing and top-quality lectures
Great lecture. Nice and slow so I can get it, and every detail is explained. This really helps me retain the knowledge when I know the how's and why's.
Your videos are saving me in my calculus class right now. I cant thank you enough for providing these videos for everyone to use.
Wow.. I wish my professor was this good at explaining. Great vid
So like others, I am taking an online class, but I just want to thank you as well... you're amazing. "follow the dy/dx". Makes SO much more sense.
Thanks you so much for making these videos.
Thank God for these videos! They're really saving me in my class. Thank you, professor.
You are an amazing teacher.
I love you professor Leonard 🙌🏼I finally get calculus and everything makes sense.
Introduction and Relevance of the Chain Rule
● [0:00]. Introduction to the Chain Rule and its importance in calculus.
○ The Chain Rule as the last main differentiation rule.
○ Applications of the Chain Rule and its use in real-world problems.
○ The three fundamental differentiation rules: Product Rule, Quotient Rule, and Chain Rule.
○ The Chain Rule as a method for deriving compositions of functions.
Motivation and Examples of the Chain Rule
● [1:19]. Review of the concept of function compositions.
● [1:26]. Example illustrating the need for the Chain Rule: derivative of (3x² - 4)^100.
○ Difficulty of deriving the expression (3x² - 4)^100 using traditional methods.
● [2:31]. The Chain Rule as an efficient solution for this type of derivative.
Explanation of the Chain Rule Through an Example
● [3:06]. Expression of (3x² - 4)^100 as a composition of functions.
● [3:40]. Method to identify the inner and outer functions in a composition.
● [4:01]. Definition of y = u^100 and u = 3x² - 4.
● [4:47]. Verification that the composition of y and u results in the original expression.
● [5:07]. Calculation of dy/du and du/dx.
○ Objective: find dy/dx.
● [7:50]. Justification of Leibniz notation for the derivative.
○ *The Chain Rule*: dy/dx = (dy/du) ⋅ (du/dx).
● [8:45]. Application of the Chain Rule to the example (3x² - 4)^100.
Interpretation and Generalization of the Chain Rule
● [12:06]. Simplification of the differentiation process using the Chain Rule.
● [13:20]. Definition of the Generalized *Power Rule*.
● [14:05]. Demonstration of the Chain Rule from the definition of compositions.
d/dx [(f(x))^n] =
y = u^n, u = f(x)
dy/du = n ⋅ u^(n-1); du/dx = d/dx [f(x)]
dy/dx = (dy/du) ⋅ (du/dx)
dy/dx = n ⋅ u^(n-1) ⋅ d/dx [f(x)]
= n ⋅ {f(x)^(n-1)} ⋅ d/dx [f(x)]
● [16:26]. *General formula for the derivative of a function raised to a power*:
○ d/dx [(f(x))^n] = n ⋅ {f(x)^(n-1)} ⋅ d/dx [f(x)]
● [18:40]. Statement in simple language of the Generalized Power Rule.
1. Multiply by the exponent n.
2. Keep the base function raised to n−1 (i.e., reduce the exponent by one).
3. Multiply by the derivative of the base function.
Examples and Applications of the Chain Rule
● [19:50]. y = (x³ - 2x + 38)⁴ find y'
○ Importance of recognizing function compositions when applying the Chain Rule.
○ Common errors when applying the Generalized Power Rule (forgetting to subtract 1 from the exponent).
○ Use of parentheses to indicate the multiplication of the inner derivative.
○ Relationship between the Generalized Power Rule and the simple Power Rule.
Additional Examples: Combination of Differentiation Rules
● [24:25]. d/dx [(x)⁶]
○ Could ypu do the general power rule?
● [26:39]. d/dx [(2x - 3)(x² - 5)³]
○ Example combining the Product Rule and the Generalized Power Rule.
○ Determining which rule to apply first: the Product Rule applies to the entire expression.
○ [29:38]. Application of the Product Rule as the main rule
● [27:28]. d/dx [(x² - 1)² / (x + 4)]
○ Identifying the necessary differentiation rules: Chain Rule (or Generalized Power Rule) and Quotient Rule.
○ Order of application: Quotient Rule first
●[28:00]. d/dx [(x² - 1)² / (x + 4)]⁵
○ Order of application: Generalized Power Rule, then Quotient Rule, and finally another Generalized Power Rule
within the Quotient Rule.
●[29:38]. d/dx [(2x - 3)(x² - 5)³]
○ Application of the Product Rule as the main rule
●[39:05]. d/dx [ √(5x² - 1)]
○ Application of the Chain Rule to derivatives involving square roots.
○ Representing square roots as fractional exponents to facilitate differentiation.
Extension of the Chain Rule to Trigonometric Functions
●[43:08]. Derivative of the function cos(x⁴) using the Chain Rule.
○ Identifying the outer function (cosine) and the inner function (x⁴). [27]
○ Applying the Chain Rule: derive the outer function, keep the inner function unchanged,
and multiply by the derivative of the inner function.
●[46:07]. *Statement of the Chain Rule for trigonometric functions*:
○ d/dx [f(g(x))] = f'(g(x)) ⋅ g'(x).
Additional Examples and Practice with the Chain Rule
● [48:22]. Practice with examples combining the Chain Rule, Product Rule, and Quotient Rule.
● [49:09]. Example d/dx [sin(4x⁵)]
○ Example of a trigonometric function with a composite argument
○ Identifying that this is a Chain Rule.
○ Recognizing the composition of functions: the outer function is sine and the inner function is 4x⁵.
○ Applying the Chain Rule: derive the outer function, keep the inner function unchanged, and multiply
by the derivative of the inner function.
● [52:15]. Example d/dx [cos²(x⁴)]
○ *Recommendation*: analyze the structure of a problem before applying differentiation rules.
○ Distinction between the general Chain Rule and the Generalized Power Rule.
○ In this context, the Generalized Power Rule refers to the Chain Rule applied to a function raised to a power.
● [1:01:08]. Example d/dx[tan(3x² - 2x)]
○ If d/dx[tan⁴(3x² - 2x)] then first general power rule
○ Identifying the Chain Rule: the outer function is tangent and the inner function is 3x² - 2x.
○ Applying the Chain Rule: derive the outer function, keep the inner function unchanged, and multiply by the
derivative of the inner function.
○ Common errors when applying the Chain Rule to trigonometric functions: forgetting to multiply by the derivative
of the argument.
● [1:06:37]. Example d/dx[(x³ + csc(x³))^(-1/2)]
○ Simplifying the expression by moving the negative exponent to the denominator.
○ Identifying the Generalized Power Rule: the outer function is raising to the power of -1/2 and the inner function
is the expression within the parentheses.
○ Applying the Generalized Power Rule: bring down the exponent, keep the internal expression unchanged, and multiply
by the derivative of the internal expression.
○ To derive the internal function, it is necessary to apply the Sum Rule, Product Rule, and Chain Rule.
● [1:19:40]. Example d/dx[(3 + x²⋅cot(x²))^(-3)]
○ Example illustrating the convenience of simplifying expressions before applying the Quotient Rule.
○ In this case, d/dx[1 / (3 + x²⋅cot(x²))^(3)], it is recommended to move the denominator to the numerator with a
negative exponent to apply the Generalized Power Rule.
○ Avoid using the Quotient Rule, as it can complicate the differentiation process.
● [1:31:20]. Exercise d/dx [(1 + cos(x²)) / (1 + sin(x²))]
○ If d/dx [(1 + cos(x²)) / (1 + sin(x²))]⁵ then apply general power rule, quotient rule and chain rule
● [1:32:10]. Exercise d/dx [x²⋅sin(3x)]
○ Product rule and then chain rule
Awesome video. My book made it impossible to comprehend and my teacher didn't do a very good job explaining it. After watching the video I now understand what I need to know. Thanks professor!
1:14:01
I had a hard time seeing this.
I had to take one more step in order to clarify it for myself that the derivative of x^3 is not going to be multiplied into the 3x^2. I did:
d/dx[x^3] + d/dx[csc(x^3)] and broke those two things up first so I could see that one derivative didnt have anything to do with the other derivative.
Im glad someone in class asked so that you could clarify that one. Very helpful videos. Thanks Prof. Leonard
You are the best teacher I ever had! Thank you so much!!!
It blew my mind when he asked “what about dy/dx” @ 8:00
Nobody had ever explained chain rule to me like that. Man is a miracle worker
I am so happy I found your videos, you don't skip steps and are very clear. Just wanted you to know that you are having a positive impact on my learning. Thanks a million from Canada.
You’re still saving lives Mr.Leonard
Thank you 😊
one of those lecture that i can keep it for long time in my heart and brain
thank you proffessor leonard you are best
WOW!!! I finally understand the chain rule! Thanks Professor!
You save lives by these videos! thank you for teaching us everything with an ease, God bless you Professor Leonard❤️
I always watch these videos as a compliment to my lectures it's nice to see these concepts from different angles
Thank you so much Prof,am from Mombasa Island in Kenya-Africa...I really appreciate your efforts ,for sure have learned alot about the rules in differentiation.may God bless you abundantly
Thanks Prof. Leonard! All you're videos are awesome - they helped me out in both math and physics
!!THANK YOU SO MUCH - شكرًا جزيلاً
your new student from Saudi Arabia
I am glad that I found such an amazing teacher in my life. Thanks for making an extra effort to let students like us get the chance to fill our bosoms with the shower of the knowledge that you give.
You honestly do not know how appreciated you are, you are a wonderful mathematician, you have cured me of so many intangible curiosities i had. You are intuitive, fun and an amazing inspiration to everyone, thank you SOOO much Professor Leonard, for making these videos, they legitimately mean a lot.
I watched 5 minuyes of this and understood chain rule more than I have in the last week of class/tutoring.
These videos are so easy to digest that I actually watched beyond what my calc professor has taught us so far, and I didn't even notice it. I checked my notes and my professor stopped at sum and difference derivatives and I'm already on the chain rule. I feel like that's truly the mark of a great teacher when you don't feel so overwhelmed you have to stop at one or two lessons.
I finally get chain rule.. I must have read the stupid text book a dozen times and watched the class lesson we are given at least three times.. Made zero sense as to what exactly I was doing.... Now I get it! Wow thanks man!
These videos guaranteed me an A in class and encouraged me to take calc 2 :) , Thank you very much
Passing Tulane calculus solely because of these videos. Professor Leonard is a hero.
Never able to learn calculus in my college, but after my college i.e vacation after last semester of college learning calculus for no reason I mean I will sure not get marks for it now, but doing it cause it is fun cause of Leonard, and yes , today I am staying awake whole night watching his video.
Thank you so much prof.. I have looked many videos on you tube and definitely you are among the best if not the best. You've made it easy to understand.
Need to keep going gotta complete all calculus by 31 december
I was taught all this in the beginning of the year (class 11 ) but not so clearly now I am getting everything clearly..
Thanks a lot
thank you professor! im taking calc online (which was a big mistake) and youve helped me a lot! more so than the professor im paying to help me learn!
Whew....those were intense, but I was able to follow them. Thanks Prof Leonard for simplifying the topic.
man id be screwed if I hadnt stumbled on these vids. The why college is explaining this leaves a lot to be desired. Thanks for saving my grade Professor Leonard!
you really helped me a lot and just saying thanks isn't enough for you❤ you owe me a lot
This free youtube series has taught me more than my 30k/year university has taught me.
Thank you so much. My professor spent like 20 minutes on this and I was so confused.
Simple and to the point. Excellent review of basic rules.
Professor You're awesome thanks for sharing the knowledge, you have helped me alot
incredibly talented professor. saved my life!
Ok so could/should you pull a "2" out of the factorization at 37:34? From 14 18 and 10
Thank you so much, sir; I have a quiz tmr and ur helping me a lot. God bless u man.
thanks prof..following your videos from Kenya..they are of great help..!!
Excellent work Professor Leonard!!!
You're the man! Awesome tutor I ever had!
best explanation and examples ever keep going
I'm working through his Calc 3 videos now but it helps to come back to these videos for a refresher on these concepts. :)
I was so into the lecture that when the kid sneezed at 1:11:47 I said Bless You. amazing lecture!!
just to let you know i answer you all the time... " from the youtube audience"...
Ha and with headphones and an iPad I look crazy for doing so. Haha..
I put my hands up when I understand! hahaha.
Thank you so much mr leonard! These lessons help me so much.
Thank you sooooooooooooooooooo much!!!!!!!!!!!!!!!!!!!!! Loving this series. Watched every video up till now and still going on :)
Thank you for a very clear explanation of the chain Rule. I believe there was a small error at 34:15 where the final d(x^2)dx should be 2 rather than 2x. But it didn't distract from the main point of the discussion so thanks again for providing such a good series on calculus. Jerry E Shepherd
+Jerry Shepherd derivative of x squared is 2x
YOU ARE SERIOUSLY AMAZING AND I CAN'T THANK YOU ENOUGH FOR YOUR VIDEOS... WOW I HOPE EVERYTHING IS GOOD IN YOUR LIFE BEST BLESSINGS!
one of the best teacher!
I have one question for the last problem where the quotient rule was used. Why did you switch from brackets to parenthesis? Couldn't you keep using brackets for the entire problem?
best lecture on chain rule
Boys are gay for Ryan Reynolds, men are gay for Professor Leonard
That was hailious " so i come with theme music" i was laughing, good to have a since of humor.
You have that user name and no videos. shame
Blows my mind that engineers had to do this all day before computers...and all those bridges and buildings are still standing!
Professor Leonard is GOLD.
You are the TRUTH. Thank you for your help. I VERY much appreciate it.
You explain these concepts so well, thank you very much!!
2024 and professor leonard is helping me get that degree
ان الله يحب إن عمل أحدكم عملاً أن يتقنه.
you are epic
You could have also factored out csc(x^2) at 1:28:50 since cot(x^2)=cos(x^2) * csc(x^2)
Great job on the videos by the way they're really helpful.
Hi Professor Leonard, just a heads up, but at 58:54 couldn't you use the sin double angle formula to further siplify the expression and end up with -4x^3(sin(2x^4))? Love your videos, and grinding through the playlists! Thank you so much!
You can't touch the argument of a trig. function. You can't bring the 2 from the outside and put it inside your argument, that changes the equation.
I think the reason why he is so built is bc he carries us with calc.
What a great professor!!