Every Professor should observe how this man teaches. THIS is what it is all about, clear, concise with the ability to convey EXACTLY what he is trying to get into the students head. Thank you professor Leonard from Las Vegas! Tim
Some people seem to think that detailed explanations are some sort of "for dummies" approach; I have always hated that description, because it seems to imply that it is only people with major learning difficulties in the subject (who I guess are the supposed "dummies") who desire clear and detailed explanations. But that is how things SHOULD be taught; if people are new at something, then they WANT hand-holding.
15 minutes into the video and I am like so that's how you do it. 5 weeks into the semester and did not understand anything until now. He is the difference between doing what you love vs doing it for a paycheck. He has passion and patience and that makes him one of a kind. you don't come across Professors like this often. Thank you Professor L.
Just got a 95% on my first calculus exam at UT Austin! The class average was a 65% and your videos are the reason I did so well. Sincerely, Professor Leonard, thank you so much. You’re the best of the best.
@@dhf335434 Haha yeah, it's really frustrating how a lot of math teachers are so goddamn dry and serious. All that grave super-stiff seriousness drives me crazy, so it's hugely refreshing that Professor Leonard has a much more relaxed and charming personality.
I know these videos are almost 10 years old at this point, But professor you really are incredible. I didn't have the strongest mathematics background knowledge but you have made everything so clear and easy to follow with some practice. You deserve all the good karma in the world.
i literally WANT to watch the next videos and i actually look forward to making my way through his playlists...this is something i have never before encountered regarding math in my life until professor leonard
thank you very much Professor Leonard....this channel is really helping me ace my math classes. Ever since i started watching the videos I became the math guru in my class and I top my class every test and exercise
The way you teach is near perfect! the ability to clearly and concisely teach the material and allow complex rules/steps/material to tie in all while making sure your students COMPREHEND is very profficient, damn near excellent. Thanks Professor Leonard! I passed my first test and midterm because of you!!!!!!!! I appreciate you
Darkenedbyshadows I'm not lying about that. It's very easy to do both or get approved. I got approved by a counselor and got approved by the head of my college math department at my local community college.
This teacher connect with the students, and he explain everything so clear, Im retired and just never understand in my college years what really was doing and passed calculus like a robot, thanks Leonard.
30:30 ... we've been doing it this whole time... mind blown. Thanks Professor Leonard, I go to a huge university and it's nice to have a teacher teach a small class. You answer the questions I have without me asking them.
You have saved my @$$! Just listening to you teach the chain rule made everything come together within the first 10 minutes. I worked on all the problems you presented before seeing the solutions and got them all correct! My teacher explains it like we've already taken calculus and should know what she's saying, it's so confusing just listening to her.. She leaves out so many parts that you take the time to teach. I have resorted to your videos first so I know what I'm doing then I go to my teachers videos to see what she's "trying" to teach. Thanks so much Professor Leonard! Keep up the awesome work!!
Great teacher, I'm so fortunate to have encountered his videos, Professor Leonard's tutorials are the best and I tip my hat. He's been my online instructor in Calculus I and will remain so until I get to Differential Equations. He's the best of the best.
i admire this guy.no jokes no bullshit no jabbering no chatting just seriousness and straight to explanation i wish all math teachers/lecturers/professors should be as him this guy should've shown all the ones across the globe whose job is to educate.
Knowing what you teach and teaching what you know are as different as chalk and cheese. Well done sir for belonging to the latter definition, you truly are inspirational. Many thanks for all the help you give those like myself who struggle with some mathematical concepts.
Arusha Braithwaite I am very envious of Merced college calculus students for having such an amazing instructor. I wonder if they realize and appreciate it themselves. I would have taken all of his classes if he taught at my college.
+Dancing Robot I like his teaching skills but I still wouldn't attend his college reason why is when I google the college it said it was a community college.
trying to teach myself this stuff and I got stuck on this. A lot of other books/videos really breeze through this without an intuitive explanation. I feel a million times more confident now, thanks so much for the videos!
I can't believe how helpful this is. My professor knows the material, he just doesn't care about his delivery. Every calculus professor should watch his videos to learn how to actually teach!
Taking Calc 1 in 2023... and this video helped me understand better than what they give you in the books or examples through the Mathlab stuff... Thanks, Professor Leonard
i seriously hate mathematics but studying calculus right now, i forgot to watch your videos to keep up with my course and boy do i want to close tabs of those tutorials and watch this 1 hour long lecture. This is how great this man is
My calculus professor(state university) just spent about 3 minutes explaining this, which is to say he did one example and moved on to derivatives of inverse trig functions. I don't know what I would do if I couldn't watch these after my actual class lectures. I'm an engineering student and I actually want to understand this material, not just pass the class. You are amazing, professor Leonard. I can't thank you enough.
Thank you. I always understood the rule of "take dy/dx whenever you see y" but I didn't know the reason for it. As everyone else has said your video series is so helpful!
Thank you for your videos! I am paying $600 for an online calc course from an accredited university where the teacher doesn't even post lectures, we are supposed to read the material and self teach a subject like this. You are the man
I'm currently enrolled in a cal 1 course at my university and today is my quiz and I didn't know anything previous week about cal and today I'm ready for my quiz just because of this professor, heads off to you, because of you I'm able to do cal otherwise I won't be able to do this course with the instructor I got.
Introduction to Implicit Differentiation ● [0:00]. Introduction to the topic: implicit differentiation. ○ Review of the explicit form of equations y = f(x): ■ Example: y = 3x² + 4 ■ Definition of explicit form: y is isolated on one side of the equation, "y" is given explicitly. ■ Explicit Form: In this form, y is expressed directly in terms of x (e.g., y = 3x² + 4). This makes differentiation straightforward as standard rules can be applied directly. ● [1:30]. Definition of implicit form: ○ Example: y + x⋅y = x ■ of he times we can make implicit function into an explicit. ▹ Solve for y ○ [2:25]. Example: x² + y² = 25 ■ y is not isolated; the variables x and y are mixed. ■ Implicit Form: Here, y is intertwined with x in the equation (e.g., x² + y² = 25). This requires treating y as a function of x and often necessitates the use of the chain rule during differentiation. ■ [4:18]. *Implicit functions can often define two different functions* ▹ x² + y² = 25 ≡ y = ±√(4 - x²) ◇ y = √(4 - x²): Represents the upper semicircle (positive). ◇ y = -√(4 - x²): Represents the lower semicircle (negative). ▹ Explicit differentiation fails when y isn't a single function, like in y = ±√(4 - x²), which represents two semicircles. Implicit differentiation handles this directly by differentiating the entire equation (x² + y² = 4) without solving for y. This allows for slopes that depend on both x and y. Definition of the Implicit Differentiation Technique ● [4:55]. Review of the concepts of explicit and implicit forms. ○ Understanding the distinction between explicit and implicit forms is crucial because it determines the method used for differentiation. In explicit differentiation, y is expressed directly in terms of x, making it straightforward to apply standard differentiation rules. In contrast, implicit differentiation is necessary when y cannot be easily isolated, requiring the application of the chain rule during differentiation. ○ In explicit differentiation, the relationship between y and x is clear and direct, allowing for straightforward differentiation. However, many real-world problems result in equations where y cannot be neatly isolated. Implicit differentiation allows us to find dy/dx without rearranging the equation, by treating y as a function of x and applying differentiation rules accordingly. ●[5:20]. Example: 2x³ + 3y³ = 9xy ○ [5:40]. It is not always possible to convert an implicit equation to an explicit form. ■ We need another method to find derivatives, instead of just assuming y equals a function of x. There has to be a different way-and there is. It's called implicit differentiation. ●[8:02]. Example: x³ + y³ = 5 ○ Example of converting implicit equations to explicit forms. ■ Some equations naturally present in implicit form cannot be easily rearranged to solve for y. Attempting to isolate y might be complex or impossible, especially with higher-degree polynomials or transcendental functions. Implicit differentiation provides a systematic way to find derivatives without needing to solve for y explicitly. ○ To solve x³ + y³ = 5, we could solve for y, subtract x³, and take the cube root, allowing us to find the derivative directly. However, the goal is to learn implicit differentiation, which lets us find the derivative without solving for y. ■ [9:33]. Key point: *You must treat y as a function of x*, because if y can be expressed in terms of x, it depends on x. Steps for Implicit Differentiation ●[9:50]. : Differentiate both sides of the equation with respect to x. ○ Remember that you are differentiating with respect to x, while y is a function of x. ○ You can differentiate term by term if the equation allows. ○ Derivative if both sides: ■ d/dx[x³ + y³] = d/dx[5] ○ When differentiating a term that includes y, apply the chain rule. ■ This requires treating y as a function of x and often necessitates the use of the chain rule during differentiation. ○ d/dx [y] is the derivative of y with respect to x, represented as dy/dx. ○ Each time you differentiate y, you must obtain a dy/dx due to the chain rule. ●[19:00]. : Solve the equation for dy/dx. Examples of Implicit Differentiation ● [23:30]. Example 1: 3y² + sin(y) = 4x⁵ ○ *Differentiate both sides of the equation*. ○ [28:52]. The key idea is to remember that you always obtain a dy/dx when differentiating y. ○ [30:50]. Implicit differentiation is simply differentiating without being able to solve for y. ■ implicit differentiation is used when you cannot (or do not) solve for y explicitly in terms of x. Instead, you differentiate both sides of an equation as they are, treating y as a function of x (i.e., y = y(x)) and applying the chain rule where necessary. ● [31:28]. Example 2: x⋅y = 1 ○ You must use the product rule when differentiating xy. ○ Differentiate both sides of the equation. ○ Obtain: y + x⋅dy/dx = 0 ○ Solve for dy/dx: dy/dx = -y/x ○ [36:00]. In implicit differentiation, you must identify when to apply the product rule or the quotient rule. Second Derivative of an Implicit Function ● [36:12]. Example: 3x² - y² = 16 ○ Find the first derivative ● [38:06]. To find the second derivative, differentiate the first derivative with respect to x. ○ Use the notation d²y/dx² for the second derivative. ○ You must use the quotient rule to differentiate the first derivative in this example. ○ When calculating the second derivative, you can *substitute dy/dx with the expression found in the first derivative*. ○ You can simplify the expression for the second derivative if possible. Geometric Interpretation of the Implicit Derivative ● [43:51]. The implicit derivative represents the slope of the curve at a given point. ● [44:22]. Example: y² - x + 1 = 0 ○ Find the derivative: dy/dx = 1/(2y) ○ Evaluate the derivative at the points (2, -1) and (2, 1) to find the slopes at those points. ○ In implicit differentiation, you don't always need both xx and y coordinates to find the slope. Whether you need both depends on the equation. If dy/dx depends on both xx and y, you'll need both coordinates. However, if y can be expressed in terms of x, knowing x may be enough. Equation of the Tangent Line to an Implicit Curve ● [47:28]. Implicit differentiation can be used to find the equation of the tangent line to a curve at a given point. ● [48:01]. Example: 4x⁴ + 8x²y² - 25x²y + 16y⁴ = 0 at the point (2, 1) ○ Differentiate the equation implicitly. ○ Solve for dy/dx. ○ Evaluate dy/dx at the point (2, 1) to obtain the slope of the tangent line. ○ Use the point-slope formula to find the equation of the tangent line. Related Rates ● [1:07:21]. Introduction to the concept of related rates: word problems involving implicit derivatives. ○ Techniques for solving related rates problems will be presented in the next session.
This is great, I'm self studying using "A First Course in Calculus" by Serge Lang and was having trouble seeing the implicit differentiation taking place, this video finally gave me that "ah ha" moment
There is a professor in Fluid Mechanics called Ihab Adam from Alexandria university of Egypt. He is the second version of you Oh my god you both are universal great❤
My Calculus teacher expects our class to learn based off of a PowerPoint per chapter, and 2 minute video clips given by the math book. This has been a huge help thanks! And he doesn't go over the PowerPoint, he just does problems in class ;(
PROF. LEONARD please I can't wait for a comeback video on your channel, I am waiting ever since I completed those polar coordinates and the whole playlist in under a month, and still. Your board in precalculus always featured " May the MA be with you" and it did with me so am here with your blockbusters ( but still craving for you to look at me in your upcoming super awesome calc. series). I am here to learn stuffs and am finding it better to be more helpful than the millions of pennies that would have gone away, if only god never allowed me to find you. Thanks to Jesus from your side and Thanks to Krishna from mine. am a 10th grader, and so my parents don't allow me to go on more than youtube , so I never found you elsewhere. Congrats for your new house and godspeed for the upcoming adventures we'll face together in maths. please, please let me digest your absence, thru u replying on at least the first line...... I'LL BE WAITING
The most satisfying feeling when I pause the time video and did the last problem fast-forwarded the video and got the answer correct! My goodness, someone nominate this guy for a Noble Peace Prize
16:08 What you do on one side, you must do on the other side: Typically you get an equation like y = 3x^2 +4 and you would take the derivative of the right hand side of the equation ** dy/dx[3x^2 +4]. However, if I understand this correctly, its really set up as: d/dx[y] = dy/dx[3x^2 +4] And so the key piece of information is hidden in there; that d/dx[y] = dy/dx
Thank you so much for taking the time to lecture this course and upload it here. Very very straight forward and clear explanation. Helped me so much with my math class!!!!
Better than most. At 13:40 tries to explain logic behind multiplying by dY/dX but does it rather mechanically. Additionally it would be helpful to develop the concept of primary Independent variable (IV), usually X and secondary IV often called "inner function" The Dependent vaiable (DV) is then what is often called the "outer function". This allows us to neatly define Implicit Function (IF) as the secondary IV or "inner function" which can only be NON-NUMERICALLY or implicitly differentiated by the primary IV (usually X). In explicit chain differentiation the secondary IV or "inner function" can be NUMERICALLY or explicitly differentiated by the primary IV (usually X).
Dear Professor Leonard (and any other boffin who may see this ;) ) P.S. Prof: I'm sorry about spamming these on your videos. I'm just in desperate need of help. I understand if you can't provide it personally - you seemingly have a lot on your plate as is! However, I am hopeful that at least somebody knows how to get around this. I am studying a Calculus I course at my university, but the work is wayyy more rigorous than how it is laid out in the textbook, or even how it is on the internet. Professor Leonard has helped me a lot in getting me to understand the basics and my marks have gone up by 5-10% ever since. But I still can't understand some concepts in the calculus context. (ie. Triangle inequality, bijection, invertible, and many others). For a better idea of what I am complaining about, here is a OneDrive link with my previous homework assignments: @t Thank you so much for anyone who may help me! Also, thank you to Professor Leonard for giving the motivation and confidence to see that I can get around this huge obstacle. I may not be around it yet, but you have at least given me the confidence and have picked me up when I was down
Weird, a Cal teacher who doesn't act like his students should already know the material
soooo true =_="
FUCKING TRUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU
this may be the most spot on comment
Justin Mabry this comment is painfully PAINFULLY TRUE!!!!!!!!!
That's the reason that I'm here in the first place!!!!
Every Professor should observe how this man teaches. THIS is what it is all about, clear, concise with the ability to convey EXACTLY what he is trying to get into the students head. Thank you professor Leonard from Las Vegas!
Tim
Some people seem to think that detailed explanations are some sort of "for dummies" approach;
I have always hated that description, because it seems to imply that it is only people with major learning difficulties in the subject (who I guess are the supposed "dummies") who desire clear and detailed explanations.
But that is how things SHOULD be taught;
if people are new at something, then they WANT hand-holding.
I am from Malaysia! this teacher is very good !!!
if i had him in 1985 for calculus, i wouldn't have failed the first time i took it
15 minutes into the video and I am like so that's how you do it. 5 weeks into the semester and did not understand anything until now. He is the difference between doing what you love vs doing it for a paycheck. He has passion and patience and that makes him one of a kind. you don't come across Professors like this often. Thank you Professor L.
Just got a 95% on my first calculus exam at UT Austin! The class average was a 65% and your videos are the reason I did so well. Sincerely, Professor Leonard, thank you so much. You’re the best of the best.
Professors should watch you so they can learn how to teach.
+monsterguy111 and how to lift
+MrSoultaker432 and be aware of the fact beauty and brains exist
and be funny at times !
But my teachers will start saying that he is foolish but acc. to me he is the best.
@@dhf335434
Haha yeah, it's really frustrating how a lot of math teachers are so goddamn dry and serious. All that grave super-stiff seriousness drives me crazy, so it's hugely refreshing that Professor Leonard has a much more relaxed and charming personality.
I know these videos are almost 10 years old at this point, But professor you really are incredible. I didn't have the strongest mathematics background knowledge but you have made everything so clear and easy to follow with some practice. You deserve all the good karma in the world.
idk what they are paying him, but professor swole needs a raise!!
Whatever it is, it's not enough.
Pretty sure he gets donations that are well deserved.
THE BEST PROFESSOR?? anyone agrees with me?
Aung Lin Hands down the best. He breaks it down into clear understandable English.
Brohh u alive ??
i literally WANT to watch the next videos and i actually look forward to making my way through his playlists...this is something i have never before encountered regarding math in my life until professor leonard
Same! His enthusiasm and clear explanations are everything. Complete opposite to my current Cal professor 😵💫
This just shows that taking your time in teaching improves the students' understanding tremendously. Unlike most professors in college. Thank you.
thank you very much Professor Leonard....this channel is really helping me ace my math classes. Ever since i started watching the videos I became the math guru in my class and I top my class every test and exercise
Awesome!! Very proud to have helped :)
The way you teach is near perfect! the ability to clearly and concisely teach the material and allow complex rules/steps/material to tie in all while making sure your students COMPREHEND is very profficient, damn near excellent. Thanks Professor Leonard! I passed my first test and midterm because of you!!!!!!!! I appreciate you
Professor Leonard puts many math instructors to shame
So fucking true bro! Im taking cal 1 and cal 2 at the sametime in college and hes better than both
mjohnson510 I don't know how you got away with taking Cal 1 and Cal 2 at same time.
Dancing Robot ikr? u kinda need one to do the other.... my bullshit meter is going off....
Darkenedbyshadows I'm not lying about that. It's very easy to do both or get approved. I got approved by a counselor and got approved by the head of my college math department at my local community college.
mjohnson510
If you are able to do Calc 2, why bother going through Calc 1? Unless you are attempting to better your marks from Calc 1.
This teacher connect with the students, and he explain everything so clear, Im retired and just never understand in my college years what really was doing and passed calculus like a robot, thanks Leonard.
I’ve used his videos all throughout college, and finally graduating in May 😂Thanks professor Leonard!
You are now my online math Professor.
Life saver! Man i wish you were my teacher hah
Pretty sure you won't ever need this in your life again haha
@@ryu4592 lol
So true
@@ryu4592lucky guy
you nailed it professor.
i dont think so any teacher or any channel explained this much detailed.
thank you
Thank you! I really appreciate that
Coronavirus made my calc class go online and it been harder for me too learn so its nice that I can come here and watch the actual lecture.
Just Started differential equations this week, been about 4 years since I took calc 1. You are a life saver thank you professor!
30:30 ... we've been doing it this whole time... mind blown. Thanks Professor Leonard, I go to a huge university and it's nice to have a teacher teach a small class. You answer the questions I have without me asking them.
THIS MAN WAS BORN TO TEACH. HE EXPLAINS THIS BETTER THAN ANYONE EVER COULD
You have saved my @$$! Just listening to you teach the chain rule made everything come together within the first 10 minutes. I worked on all the problems you presented before seeing the solutions and got them all correct! My teacher explains it like we've already taken calculus and should know what she's saying, it's so confusing just listening to her.. She leaves out so many parts that you take the time to teach. I have resorted to your videos first so I know what I'm doing then I go to my teachers videos to see what she's "trying" to teach. Thanks so much Professor Leonard! Keep up the awesome work!!
You should do a video on how to lift.
For real though. PL inspired me to get in shape
was going to like this comment but it was at 69 likes
U alive buddy ??
Math Professors Hate HIm.
Great teacher, I'm so fortunate to have encountered his videos, Professor Leonard's tutorials are the best and I tip my hat. He's been my online instructor in Calculus I and will remain so until I get to Differential Equations. He's the best of the best.
i admire this guy.no jokes no bullshit no jabbering no chatting just seriousness and straight to explanation i wish all math teachers/lecturers/professors should be as him this guy should've shown all the ones across the globe whose job is to educate.
I got and A in my math test!!!! Thank you Professor Leonard, you've helped me improve my grades!!!
Holy shit man you are the superhero of math professors man! Thank you sooooo much!!!! Never have I ever wanted to watch calc videos.
Knowing what you teach and teaching what you know are as different as chalk and cheese. Well done sir for belonging to the latter definition, you truly are inspirational. Many thanks for all the help you give those like myself who struggle with some mathematical concepts.
Once upon a time people left their homes and traveled thousands of miles to find the best school and the best teacher. Now .....
It's like the Issac Asimov short story, "The Fun they Had"
10 years later.....we still here 😂
He is sent from heaven 🙌
Arusha Braithwaite I am very envious of Merced college calculus students for having such an amazing instructor. I wonder if they realize and appreciate it themselves. I would have taken all of his classes if he taught at my college.
+Dancing Robot I like his teaching skills but I still wouldn't attend his college reason why is when I google the college it said it was a community college.
Master of Math You go to community college to get your gen eds at a decent price, then transfer to your uni of choice. Duh.
@@belldanime that wasnt what he/she was saying you ape
trying to teach myself this stuff and I got stuck on this. A lot of other books/videos really breeze through this without an intuitive explanation. I feel a million times more confident now, thanks so much for the videos!
I can't believe how helpful this is. My professor knows the material, he just doesn't care about his delivery. Every calculus professor should watch his videos to learn how to actually teach!
Taking Calc 1 in 2023... and this video helped me understand better than what they give you in the books or examples through the Mathlab stuff... Thanks, Professor Leonard
This guy did a pushup everytime he solved a math problem
i seriously hate mathematics but studying calculus right now, i forgot to watch your videos to keep up with my course and boy do i want to close tabs of those tutorials and watch this 1 hour long lecture. This is how great this man is
My calculus professor(state university) just spent about 3 minutes explaining this, which is to say he did one example and moved on to derivatives of inverse trig functions. I don't know what I would do if I couldn't watch these after my actual class lectures. I'm an engineering student and I actually want to understand this material, not just pass the class. You are amazing, professor Leonard. I can't thank you enough.
Thank you Professor! You're amazing. I found myself relaxed, satiated and drenched in gratitude by the end of the lecture.
You have no idea how much this video saved my life. :) Thank you! You are a great teacher!
Thank you. I always understood the rule of "take dy/dx whenever you see y" but I didn't know the reason for it. As everyone else has said your video series is so helpful!
This man is the plug when it comes to calc. no further comments needed.
I am so glad i came across your youtube channel! I never thought Calculus could be such a walk in the park :) A+, here I come!!!
I have class more than 3 hours and I don't understand nothing but with you I feel like a teacher of calculus
Thank you for your videos! I am paying $600 for an online calc course from an accredited university where the teacher doesn't even post lectures, we are supposed to read the material and self teach a subject like this. You are the man
I'm currently enrolled in a cal 1 course at my university and today is my quiz and I didn't know anything previous week about cal and today I'm ready for my quiz just because of this professor, heads off to you, because of you I'm able to do cal otherwise I won't be able to do this course with the instructor I got.
Professor Leonard + 3blue1brown = Knowledge + Wisdom = :D
Best ever Math teacher!!!!! I wish you were my professor.
Introduction to Implicit Differentiation
● [0:00]. Introduction to the topic: implicit differentiation.
○ Review of the explicit form of equations y = f(x):
■ Example: y = 3x² + 4
■ Definition of explicit form: y is isolated on one side of the equation, "y" is given explicitly.
■ Explicit Form: In this form, y is expressed directly in terms of x (e.g., y = 3x² + 4).
This makes differentiation straightforward as standard rules can be applied directly.
● [1:30]. Definition of implicit form:
○ Example: y + x⋅y = x
■ of he times we can make implicit function into an explicit.
▹ Solve for y
○ [2:25]. Example: x² + y² = 25
■ y is not isolated; the variables x and y are mixed.
■ Implicit Form: Here, y is intertwined with x in the equation (e.g., x² + y² = 25).
This requires treating y as a function of x and often necessitates the use of the
chain rule during differentiation.
■ [4:18]. *Implicit functions can often define two different functions*
▹ x² + y² = 25 ≡ y = ±√(4 - x²)
◇ y = √(4 - x²): Represents the upper semicircle (positive).
◇ y = -√(4 - x²): Represents the lower semicircle (negative).
▹ Explicit differentiation fails when y isn't a single function, like in y = ±√(4 - x²), which represents
two semicircles. Implicit differentiation handles this directly by differentiating the entire
equation (x² + y² = 4) without solving for y. This allows for slopes that depend on both x and y.
Definition of the Implicit Differentiation Technique
● [4:55]. Review of the concepts of explicit and implicit forms.
○ Understanding the distinction between explicit and implicit forms is crucial because it determines
the method used for differentiation. In explicit differentiation, y is expressed directly in terms of x,
making it straightforward to apply standard differentiation rules. In contrast, implicit differentiation
is necessary when y cannot be easily isolated, requiring the application of the chain rule during differentiation.
○ In explicit differentiation, the relationship between y and x is clear and direct, allowing for straightforward
differentiation. However, many real-world problems result in equations where y cannot be neatly isolated. Implicit
differentiation allows us to find dy/dx without rearranging the equation, by treating y as a function of x and
applying differentiation rules accordingly.
●[5:20]. Example: 2x³ + 3y³ = 9xy
○ [5:40]. It is not always possible to convert an implicit equation to an explicit form.
■ We need another method to find derivatives, instead of just assuming y equals a function of x.
There has to be a different way-and there is. It's called implicit differentiation.
●[8:02]. Example: x³ + y³ = 5
○ Example of converting implicit equations to explicit forms.
■ Some equations naturally present in implicit form cannot be easily rearranged to solve for y. Attempting
to isolate y might be complex or impossible, especially with higher-degree polynomials or transcendental
functions. Implicit differentiation provides a systematic way to find derivatives without needing to solve for y explicitly.
○ To solve x³ + y³ = 5, we could solve for y, subtract x³, and take the cube root, allowing us to find
the derivative directly. However, the goal is to learn implicit differentiation, which lets us find the
derivative without solving for y.
■ [9:33]. Key point: *You must treat y as a function of x*, because if y can be expressed in terms of x,
it depends on x.
Steps for Implicit Differentiation
●[9:50]. : Differentiate both sides of the equation with respect to x.
○ Remember that you are differentiating with respect to x, while y is a function of x.
○ You can differentiate term by term if the equation allows.
○ Derivative if both sides:
■ d/dx[x³ + y³] = d/dx[5]
○ When differentiating a term that includes y, apply the chain rule.
■ This requires treating y as a function of x and often necessitates the use of the
chain rule during differentiation.
○ d/dx [y] is the derivative of y with respect to x, represented as dy/dx.
○ Each time you differentiate y, you must obtain a dy/dx due to the chain rule.
●[19:00]. : Solve the equation for dy/dx.
Examples of Implicit Differentiation
● [23:30]. Example 1: 3y² + sin(y) = 4x⁵
○ *Differentiate both sides of the equation*.
○ [28:52]. The key idea is to remember that you always obtain a dy/dx when differentiating y.
○ [30:50]. Implicit differentiation is simply differentiating without being able to solve for y.
■ implicit differentiation is used when you cannot (or do not) solve for y explicitly in terms of x.
Instead, you differentiate both sides of an equation as they are, treating y as a function of x
(i.e., y = y(x)) and applying the chain rule where necessary.
● [31:28]. Example 2: x⋅y = 1
○ You must use the product rule when differentiating xy.
○ Differentiate both sides of the equation.
○ Obtain: y + x⋅dy/dx = 0
○ Solve for dy/dx: dy/dx = -y/x
○ [36:00]. In implicit differentiation, you must identify when to apply the product rule or the quotient rule.
Second Derivative of an Implicit Function
● [36:12]. Example: 3x² - y² = 16
○ Find the first derivative
● [38:06]. To find the second derivative, differentiate the first derivative with respect to x.
○ Use the notation d²y/dx² for the second derivative.
○ You must use the quotient rule to differentiate the first derivative in this example.
○ When calculating the second derivative, you can *substitute dy/dx with the expression found in the first derivative*.
○ You can simplify the expression for the second derivative if possible.
Geometric Interpretation of the Implicit Derivative
● [43:51]. The implicit derivative represents the slope of the curve at a given point.
● [44:22]. Example: y² - x + 1 = 0
○ Find the derivative: dy/dx = 1/(2y)
○ Evaluate the derivative at the points (2, -1) and (2, 1) to find the slopes at those points.
○ In implicit differentiation, you don't always need both xx and y coordinates to find the slope.
Whether you need both depends on the equation. If dy/dx depends on both xx and y, you'll need
both coordinates. However, if y can be expressed in terms of x, knowing x may be enough.
Equation of the Tangent Line to an Implicit Curve
● [47:28]. Implicit differentiation can be used to find the equation of the tangent line to a curve at a given point.
● [48:01]. Example: 4x⁴ + 8x²y² - 25x²y + 16y⁴ = 0 at the point (2, 1)
○ Differentiate the equation implicitly.
○ Solve for dy/dx.
○ Evaluate dy/dx at the point (2, 1) to obtain the slope of the tangent line.
○ Use the point-slope formula to find the equation of the tangent line.
Related Rates
● [1:07:21]. Introduction to the concept of related rates: word problems involving implicit derivatives.
○ Techniques for solving related rates problems will be presented in the next session.
This is a fabulous explanation. I was stuck on how this worked even half way through calculus 2, but I found the answer!
This is my second time taking a calculus 1 course and so far you have helped me far better than my actual professor here in school. Thank you.
This is great, I'm self studying using "A First Course in Calculus" by Serge Lang and was having trouble seeing the implicit differentiation taking place, this video finally gave me that "ah ha" moment
I would never EVER miss his class unless I died... he's understandable AND smokin' HOT
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My Calculus teacher expects our class to learn based off of a PowerPoint per chapter, and 2 minute video clips given by the math book. This has been a huge help thanks! And he doesn't go over the PowerPoint, he just does problems in class ;(
why did I find this 2 days before my calculus exam? ;( ;( Very well done, thank you Prof. Leonard
PROF. LEONARD please I can't wait for a comeback video on your channel, I am waiting ever since I completed those polar coordinates and the whole playlist in under a month, and still. Your board in precalculus always featured " May the MA be with you" and it did with me so am here with your blockbusters ( but still craving for you to look at me in your upcoming super awesome calc. series). I am here to learn stuffs and am finding it better to be more helpful than the millions of pennies that would have gone away, if only god never allowed me to find you. Thanks to Jesus from your side and Thanks to Krishna from mine.
am a 10th grader, and so my parents don't allow me to go on more than youtube , so I never found you elsewhere.
Congrats for your new house and godspeed for the upcoming adventures we'll face together in maths.
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I'LL BE WAITING
You are the BEST math teacher I ever had !!!!
Thank you Prof.
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What I really admire is how calculations are well organized....if only my math teachers (in the past) could insisted on that ....
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Never had someone explain it to me like this you're right it is cool and it makes so much sense thanks
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I'm taking Differential equation and he said that we should see this lecture before we continue,
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This guy is seriously gifted, some how I can finally understand calculus.
i recommended your channel to my calc professors, told them they could learn a thing or two
I appreciate your way of teaching sir, highly recommend for my calculus professor
12:10 - 12:15 that faint "HA" after Prof. Leonards joke, idk why but it made me crack up. BTW thank you for these videos!
I never would have thought that I'll willing watch a hour long Calculus lecture fulling. But thank you
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The most satisfying feeling when I pause the time video and did the last problem fast-forwarded the video and got the answer correct! My goodness, someone nominate this guy for a Noble Peace Prize
You're actually so goated bro, understood everything w one video.
I hope he makes a return to RUclips. The man is indispensable.
Thanks professor, you're the reason I'm passing my class.
16:08
What you do on one side, you must do on the other side:
Typically you get an equation like y = 3x^2 +4 and you would take the derivative of the right hand side of the equation ** dy/dx[3x^2 +4]. However, if I understand this correctly, its really set up as:
d/dx[y] = dy/dx[3x^2 +4]
And so the key piece of information is hidden in there; that d/dx[y] = dy/dx
Best teacher ever!!!!!!🔥
Thank you so much for taking the time to lecture this course and upload it here. Very very straight forward and clear explanation. Helped me so much with my math class!!!!
I'm pretty sure my calculus teacher uses these videos as lessons guides lol. Thanks for the help, way better explanations
My teacher threw the book at us and has no lectures or lessons so thank you for these videos it’s saving us
All these months I though I was a dumb kid who wasn't fit for doing math or calculus for that matter, it really was this easy...thanks prof!
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I truly learn so much from all your videos. Thank you for uploading them, I hope you keep uploading them.
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dawg im sleep deprived tryna solve that last problem and im seeing stars LMAOAOAO
life saver. still watching in 2021! helped so much, God bless
Your lectures are extremely helpful!! Thanks for uploading!
Loving the way you teach
For Halloween, you should dress like a "Lifesaver" ;-)
The chain rule. Yes. Got it. You go into the ZEN of it. Beautiful Job.
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Great videos man! My prof is an old guy who writes in cursive and has terrible handwriting. You've helped me so much.
Are you talking about talman
Thanks and loves from Turkiye!
His teaching is so calm and relaxing
This guy could knock me off with one punch or with differential calculus 💀
This guy is a better teacher than my professor right now. They went straight to 2.8 which makes me ask how long is that damn class lol
Better than most. At 13:40 tries to explain logic behind multiplying by dY/dX but does it rather mechanically.
Additionally it would be helpful to develop the concept of primary Independent variable (IV), usually X and secondary IV often called "inner function" The Dependent vaiable (DV) is then what is often called the "outer function". This allows us to neatly define Implicit Function (IF) as the secondary IV or "inner function" which can only be NON-NUMERICALLY or implicitly differentiated by the primary IV (usually X). In explicit chain differentiation the secondary IV or "inner function" can be NUMERICALLY or explicitly differentiated by the primary IV (usually X).
Dear Professor Leonard (and any other boffin who may see this ;) )
P.S. Prof: I'm sorry about spamming these on your videos. I'm just in desperate need of help. I understand if you can't provide it personally - you seemingly have a lot on your plate as is! However, I am hopeful that at least somebody knows how to get around this.
I am studying a Calculus I course at my university, but the work is wayyy more rigorous than how it is laid out in the textbook, or even how it is on the internet. Professor Leonard has helped me a lot in getting me to understand the basics and my marks have gone up by 5-10% ever since. But I still can't understand some concepts in the calculus context. (ie. Triangle inequality, bijection, invertible, and many others). For a better idea of what I am complaining about, here is a OneDrive link with my previous homework assignments: @t
Thank you so much for anyone who may help me! Also, thank you to Professor Leonard for giving the motivation and confidence to see that I can get around this huge obstacle. I may not be around it yet, but you have at least given me the confidence and have picked me up when I was down