How to Do Implicit Differentiation (NancyPi)

MIT grad shows how to do implicit differentiation to find dy/dx (Calculus). To skip ahead: 1) For a BASIC example using the POWER RULE, skip to time 3:57. 2) For an example that uses a TRIG FUNCTION and the PRODUCT RULE, skip to time 7:20. Nancy formerly of MathBFF explains the steps.
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What is implicit differentiation? Up until now, most functions you've had to differentiate have probably been written EXPLICITLY as a function of x, such as y = x^2. However, if a function y is written IMPLICITLY as a function of x, such as x^2 + y^2 = 9, you will need to use implicit differentiation to find the derivative dy/dx. The only difference is that any time you take the derivative of y, you must also multiply by dy/dx. The reason is that y is dependent on x. You can think of y as containing some x expression inside it, so
implicit differentiation is basically a special instance of the chain rule in which you must take the outside derivative but also multiply by the inside derivative, which is dy/dx in this case.
Here are the steps to doing implicit differentiation to find DY/DX:
1) TAKE THE DERIVATIVE OF BOTH SIDES, MULTIPLYING BY DY/DX every time you take the derivative of a Y: The first step is to take the derivative of both sides of the equation, with respect to x, but to attach a dy/dx if you ever take the derivative of y. For instance, to implicitly differentiate the equation x^2 + y^2 = 9, take the derivative of both sides with respect to x. On the left, the derivative of the x^2 term is just 2x. To differentiate the y^2 term, in this case first use the power rule to get 2y, and THEN, because y is dependent on x, you must multiply the term by dy/dx so that you have y^2 times dy/dx. Don't forget to differentiate the right side of the original equation as well, which was the constant 9, so the derivative is just 0. Your new, differentiated equation is then 2x plus y^2 dy/dx = 0.
2) GET DY/DX ALONE ON ONE SIDE: The third step is to solve for dy/dx, or in other words, to get dy/dx alone on one side of the equation. Next, DISTRIBUTE AND EXPAND BOTH SIDES if necessary distributing or opening up any terms that have parentheses. Then, if dy/dx appears in more than one term, get those terms together on one side and use the SIMPLIFICATION TRICK OF FACTORING out dy/dx from those terms so that it then appears only once on that side. Finally, DIVIDE OUT any factor that is currently multiplied by the dy/dx so that you are left with dy/dx alone on one side and an equation that looks like dy/dx = some other expression.
NOTE: Only multiply by dy/dx if you are taking the derivative of y. In more challenging examples like the second example in this video, you do not need to attach a dy/dx if you are just multiplying by y. For instance, when differentiating a term like xy, using the product rule gives you x times dy/dx plus y. Notice that in the second term from the product rule, it was not necessary to attach a dy/dx to the y, since you were not taking the derivative of y in that term.
For more of my calculus and math videos, check out: nancypi.com

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Not so great for the video but she teaches well

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1.5 speed fixes that issue

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IK I LITERALLY FELL ASLEEP NOT JOKING JCNDKFJF

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Nice choice of words.

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Nancy > my calculus professors at uni. Thank you for doing these.

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facts!!

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i gotta do this stuff for my A levels too man

[@jodybrai]

14:08 "I just did a bunch of calculus, and it's not my job" CLASSIC

[@sunlin894]

What does she mean"It's not my job"????. Yet, I understand Impl. Diff.

[@jodybrai]

@@sunlin894 She means that teaching maths on KZbin is just a hobby, not what she does for a living.

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@Chemist Me I think she's a physicist, actually. I feel like she has a video talking about her profession.

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@Chemist Me I know this might be radical to some guys, but she is a well educated woman who has a grand career ahead of her, an MIT grad as well as a KZbin phenom.

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online classes is hard without proper teaching, thank you Nancy! i love how you explain every step of the way no matter how obvious it is because sometimes they can be missed out.

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OMG, I agree. I was dying inside during my first lesson of this in my online class, KZbin suggested this and it's magic! Thank you Nancy!

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right!

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U try having a teacher that barely speaks English, try and record videos with a cheap mic, and literally only reading off notes he made, he does nothing else lmao he just reads notes

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Math has never been so calming. The explanation made a lot of sense thanks!

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why hasn't anyone acknowledged the fact that Nancy is writing inside out

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its mirrored after

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@@Martin-X I think he meant that she's writing away from her body

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Right, she's writing to her right which would go away from her body. Then it's flipped

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@@Martin-X She's writing right to left ..

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@@trevorstarr5218 yes but if you mirror the image it becomes right to left

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Nancy. I'm a retired engineer and just love studying math and science as a hobby. Your videos get to the point, are clear, concise and packed with information. You have increased my understanding of Calculus. Also love the way you fast forward writing out the details. A great service to those who are studying Calculus. Thank You!

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I got an A in my Trig class because your videos helped me so much ! I'm taking calculus now and I'm so happy to see this!!! THANK YOU

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I'm in my 50's now and i've begun to realise that the difficulty in maths arises from failing to remember all the little rules rather than a failure to understand. Nothing in this video was difficult to understand, I would however repeatedly trip up over every little quirk that was highlighted! 😂😂 Thanks Nancy!

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I had this same realization a little while ago. If you just remember the rules you'll do well

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I originally took calculus 40+ years ago. At 71 years old I started reviewing my calculus so I could learn advance electronics. The book I am using is somewhat confusing and it takes several days of going over what they are trying to explain. Now I came across your videos and the explanations are more clear and easy to understand. Thank you for the help, Nancy. I am looking forward to more advanced topics.

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Nancy. Thanks for taking the time to take your time. The biggest problem with seated lectures is that the teacher knows the material so well that they go through it really fast. I never have time to fully comprehend what's going on. Going slow and explaining your steps is so incredibly helpful!

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You’re totally right!!

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Absolutely True!!!

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Honestly, much of what I've seen from your videos explain and teach calculus better and more concisely than a lot of the instruction I've gotten from college calculus professors. You're a life-saver, Nancy!

[@gabytorres1848]

i watched the khan academy videos like 5 times and I watched this one once and I understood thank you nancy!

[@victoriananthavong8809]

Miss NancyPi, thank you so much for your details! When you reminded us the simple mistakes people make, and when you explained your perspective on this subject. It feels like the chain rule’s long lost sibling.

[@sidewinderenigma]

I'm doing Calculus online through WGU, and your videos have been a major game changer in how to understand this vs just reading about it. Thank you!

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Coming back after a year off math to find Math BFF gone was a sinker, you and Khan got me through my first two years distance learning. Thankfully I found this channel and get instant revision in your familiar cool, calm manner and slightly downward sloping, backward handwriting! Please keep up the good work Nancy, for us non-math built people, you are a God send.

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When the teacher covered it, it made no sense and seemed a beast of a concept! Now thanks to you it is a small kitten! Thank you!!!!

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Thank you so much for the well constructed video and patient dialogue that walks us through each step! This is very appreciated, you're a great internet teacher!

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Nicely done, NancyPi. I like that you add points where you see some students get tripped up. I tutor math and will be sharing your videos with students! Thanks!

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OMG! Thank you so much, you're such a lifesaver. You went into detail explaining each step as any teacher should. My math professor is horrible at explaining how she do things. She just teaches us as if were already supposed to know the steps.

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Lucid explanation, and great examples!! Watched this a few months ago when helping my kiddo with calculus homework (I have 4 years of calculus classes... 25 years ago and forgot most of the mechanics.) Well, helping her study for AP exam yesterday, implicit differentiation problem came up that she couldn't solve... NancyPi to the rescue again!

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Welcome back!. Gonna use this video later in this semester thank you 👌🏻

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I'm so glad you're doing so many more topics!

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You saved me, I have a teacher who I never understand a single word from her and mathematics is my major.. thank you so much, wish you all the happiness !!

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I started watching your video sets on calc just before I started a course on it. Very helpful and easy to understand. Thanks a bunch!

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Hi Nancy, I hope you read this. You inspire a lot of us, and I hope you could soon make other videos for the Integrals, Advanced Math, and all of those sorts.. You rock! :)

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I honestly can't thank you enough Nancy. my professor is a small smelly balding man who shows his butt crack every time he tries to reach for the top of the white board, he coughs every 30 seconds (he's a heavy smoker), and he's all over the place, literally making it up as he goes. Your videos are pleasing in every way... perfect audio, great video quality, great examples and a clear purpose to everything you say, do and show! You make calculus stupid easy, because it is. Everyone who thinks math is hard either didn't try hard enough or had a poor instructor.

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Girl, you rock. I would easily pay to have more of your content about calculus. Most of the classes or books I've been reading overcomplicate much of the content. You teach it using simple terms and language, making it easier to understand. Literally turning water into wine. Thank you

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Love your videos! I am not the biggest fan of calc since some parts weren't explained as well and were confusing. Watching your videos, I find that I can grasp what is happening much easier and it makes sense. Thank you!

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Great videos always! Super helpful as a Calc 1 student to see other ways of thinking about and doing derivatives, limits, etc! :)

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Yes

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Nancy you are my saviour. Thanks to you, I will get through my master's degree

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Thank you, Nancy, for helping me study, and thank you for helping my professor teach. She actually thought your teaching style was wonderful that she listed your video as an added resource to her lecture and text videos. Keep going because I think there are Professors that notice your teaching style is pleasant, so it is possible that a professor recommend that you become part of a university as I professor. I hope that you become a professor, so that you can teach while experiencing benefits that come with being a professor.

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Thank you Nancy so much, I had to stop the video and write this comment only to thank you before I finish watching the video. This was something very difficult for me to wrap my head around it. THANKS MILLIONS. YOU DO AMAZING TUTS.

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I am struggling with BC, and your video helped me a great deal!

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I admit NancyPi's math skills are superb.....BUT IS NO ONE GONNA SAY HOW AMAZING IT IS THAT SHE IS WRITING BACKWARDS AND FACING US???? THAT IS TRUE SKILL.

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Yaaa

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Right she write in amazing formm great ...How she write like this ??🤔🤔

[@lilywaterman5717]

I forgot about the product rule and your video is the only implicit differentiation video that mentions it! Thank you for saving my calc grade!!

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Thank you so much for this Nancy. Being an ODeL student is hard. So people like you always make life a little easier

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This is the first time I have seen one of your videos Nancy. I very rarely comment on any videos, but I have to say that I think you are simply amazing! I am sure it must take so much concentration effort, being able to write everything backwards, while still keeping it all together and explaining everything in such a clear and calm manner AND never forgetting to switch between the two colors (every single time!). I have subscribed and am sure to watch your other videos as well as soon as they become relevant during my studies, Thank you so much for preparing and uploading this very professional video, much appreciated!

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so im not crazy, how do you even do that and keep it together. writing mirrored is a huge skill

[@NanoTellez]

She's not writing backwards. It's a simple post-editing trick to flip the image. Lots of people doing videos with a clear board these days, so it's a fairly well-known process. Other than that, you're spot on that her video was extremely well-done.

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@@NanoTellez Thanks for making me feel like a fool 😂 I really didn't think about the option.. So Nancy is also not left handed 😉

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I like your half thinking/half explaining face--the rhetorical "does that make sense?" to the camera haha. Good job Nancy :)

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Thank you Nancy!! I mainly been struggling with slope of tangent curve problems using implicit differentation to plug in the coordinates to. I got a final exam for calculus 1 saturday I am reviewing topics! I love your videos

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I've been struggling with implicit differentiation for days....this video made it perfectly clear to me. Thanks so much!

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id rather pay you to teach this than my university. im literally learning NOTHING from the book or professor!! you should have some merchandise to sell or something.

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you're not alone

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You're not alone

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ok simp

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@@braz1080 i make a comment about how i would rather pay someone who does a good job at teaching a subject matter vs paying my university or my professor who doesnt help and you think im a simp? whatever troll. you are watching the video too because you are taking the same class and need the help.

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defending yourself like a true simp

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I cant even explain how helpful your videos are!!! I was having a really hard time in Calc this year doing online school. Thank you so much you basically taught me the whole first semester😭😭😭

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Hi

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Nancy you are so amazing at teaching math and I'm happy I found your channel because your videos are so clear and helpful. Thank you so much!🙏🙏🙏

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Saw this tuesday for the first time and panicked, read the whole chapter and had no clue. Just watched this for 14 minutes and it's clear as day. Thank you so much!

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Wow Nancy you explained what my math teacher couldn't thanks a lot

[@rohanthomas5434]

This is super amazing. I was actually struggling in Implicit Differentiation.

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You are the FIRST person that explained what the heck this all means. I have seen the light! Thank you!

[@jeffreychavey4161]

Your rationale for why and when to use dy/dx is something other instructors aren’t so good at. Thanks for the great video

[@aayushpashine9820]

Great explanation!!! Thanks

[@hamzini0]

Thank you so much for explaining it very slowly as if you know you need to teach it to someone dumb like me . which is a good thing . helped a lot!

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Great job explaining this. I have a calc midterm today and needed to learn this unit quickly. Thank you so much!

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thank you so much you are the only who is thorough and clear enough. you helped me so much!

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THANK YOU Nancy

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Hi I really like the way you explain. I wish you could make a full course on calculus 1,2 & 3 with quizes and all.

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Wouldn’t that be nice 😂

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Fantastic. I love how simple it is when you talk about it❤

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Thank you is not enough to say, You really helped me in most of the things that I need. Thank you Nancy

[@Hamiimarie]

Hello. Just wanted to say your videos are helping me a lot. I’m really struggling in calculus, and your videos help. I was wondering, if you can maybe do a video on particle motion and optimization? Thanks

[@kristym7149]

Watching this at 10:30pm the night before my 8am AP Calc exam

[@DorotheaDw]

Honestly, just wanted to say THANK YOU! I never understood implicit differentiation from a conceptual standpoint until now! I've always thought it was just something I had to memorize, but this was the first time I understood it intuitively!

[@zack_120]

Thanks Nancy for the crystal clear presentation. For the example x^2+y^2=36, one would get the same result by rearranging the function as y=sqrt(36-x^2) and take its derivative as usual, non-implicitly because y is singled out on the left. The result is dy/dx=(-x/sqrt(36-x^2), or. -x/y.

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You are so helpful. Thanks.

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Your channel is so under rated! You explain things so well! I watched your Q&A btw! I am on the East Coast too and I loveeeeeeeeeeeeeeee pasta!! lol

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Mackenzie Costner nhi degi

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Thank you!!!!!!! i have been trying to understand this for weeks and you explained it great

[@milxb00kpro0wn3r]

Now I understand this concept. I did not know y^1 = dy/dx and that is another reason I was getting super confused on this topic. Amazing video. Thank you. It was really helpful.

[@readjordan2257]

That cleared things right up for me. The thing that clicked the most is how Y represents an X value indirectly, and im like OH Duh!

[@misan2002]

yeah same for me

[@rushiljain9423]

Yeah when you she said to take the derivative with respect to "x", meaning just do the power rule, so I am naturally associating the power rule to finding for x first. And then just equal it to dy/dx because we are being "racist" towards y lol

[@3694omz]

My goodness Nancy. You teach way better than KhanAcademy. Thank you so much ❤

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Really helpful. Clear explanation. Stress to the point. Fantastic.

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I already know how to do this but I needed to destress and your voice is so calming

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I lost it when she said "y-detector" lol

[@vaiebhavpatil2340]

When?

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*will be amusing herself with this during the final on Monday* (in buzz lightyear voice) 'you don't wanna be in the way when my y detector goes off... y'

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@@vaiebhavpatil2340 7:59

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😂🤣💀

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lmfaooo me too

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thank you uploading your videos despite all those creepy morons in the comments!

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I know right

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Yeah, seeing those weirdos making creepy statements makes me feel uncomfortable. I just want to understand calculus lol.

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I would just like to thank you for all of your videos they have helped me understand math a lot better know

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This makes so much sense now... Thank you Nancy!

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if nancypi made programming videos: nancypy

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Thank you!

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Thank you nancy. For your help in learning the derivatives

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My best math teaching channel yet.

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I fell in love every time she said dy/dx

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Baklang tooo

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I miss you. Please do complex numbers

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Nancy, i must admit your videos are very good. you really have helped me a lot today. Thank You.

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Awesome explanation! So clear and thoughtfully delivered. Thank you!

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i LOVE this girl! Thank you, you’re my savior lol 😂

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7:17 Creepy :O Btw, you make things look very simple lol xD Wish I could do like a pro in exams.

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Nancy, You are a gem!! Are you born again? Bless you a Thousand times

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Stucked on a question for so long, ur video helped a lot.Thanks a lot Maam.

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OMG it was to factor out the dy/dx. Of course. Now I feel stupid after staring at this for 20 minutes. Nancy the witch! haha you and your sorcery. xD

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thank you for sharing your intelligence, Nancy.

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Thanks Nancy I'd got lots of help from you while I've been studying maths at the University few years ago

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This was incredibly helpful. Thank you so much for making this simpler.

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Please Teacher Nancy explain absolute maxima and minima I’ll be thankful 😭🙏🏻🙏🏻🙏🏻

[@teamgalaxybreaker]

the max and min are where the graph of the function reached either the highest or lowest point possible, for example, f(x)=x^2 has a minimum of (0,0) but no maximum because both sides continuously approach positive infinity

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thanks nancy

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Hi ma'am Nancy your the reason why I always pass my exam in Differential Calculus. Thank you always😊

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I'm in a 4 week intersession calculus class and last time I studied algebra was 10 years ago. Your videos makes it so much easier to understand, the teacher explains based in algebra which has its value but it's confusing to me. You speak in terms of calculus and that makes it so much clearer for me to understand. Thanks for the easy to follow explanations

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Hi, i don't know how often you make videos a week or month but I have my second calc test next week. May you do a video on related rates?

[@masterbitter7688]

I Ned that too also optimization

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Same

[@cliffordwilliams9597]

Example: Implicit differentiation of 3x^3 + 5y^2 = 15: 3x^3 + 5y^2 = 15 (The differentiation is implied)

[@a.guzman1512]

Your calm, clear, step by step explanations helps me calm down and actually pay attention and not worry about missing something. Some instructors talk way to damn fast. Thank you.

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Thanks so much! I was really struggling to understand this concept and you've really helped me.