Derivatives... What? (NancyPi)

MIT grad shows the DEFINITION of the derivative and how to FIND the derivative using that limit definition. To skip ahead: 1) For what the derivative MEANS, skip to 0:23. 2) For the SLOPE OF THE SECANT line, skip to 2:53. 3) For the SECANT BECOMING THE TANGENT and DEFINITION OF THE DERIVATIVE, skip to 5:54. 4) For how to FIND THE DERIVATIVE USING THE (LIMIT) DEFINITION of the derivative, skip to 10:28. Nancy formerly of MathBFF explains the steps.
For my video on the shorter, faster DERIVATIVE RULES for how to take a derivative, jump to: • Derivatives... How? (N...
Follow Nancy on Instagram: / nancypi
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INTRODUCTION to derivatives:
1) WHAT IS THE DERIVATIVE? It's a function that tells you the slope (of the line tangent to the curve) at every point. Another way to think of it is that the derivative gives you the rate of change at any instant (the "instantaneous rate of change" at each point). You can find the derivative either with the proper DEFINITION OF THE DERIVATIVE ("by the limit process") or the faster, simpler way with the shortcut derivative rules such as the Power Rule, Product Rule, Quotient Rule, and Chain Rule. This video shows the first way, with the DEFINITION of the derivative, and how to use it to calculate the derivative.
2) SLOPE OF SECANT: To make the definition of the derivative, we can start with the slope of the straight secant line through a point x and some other point nearby, h distance away horizontally. If we label the two points and use the slope formula to write an expression for the slope of this secant line, the expression you get is [f(x+h) - f(x)] / h, which is also known as the DIFFERENCE QUOTIENT.
3) DEFINITION OF DERIVATIVE: That straight line slope we just found is actually a decent ESTIMATE for the slope at x, but it's not really the slope at x. We can get the EXACT slope of the tangent line at x by closing in on x and narrowing h to zero, by taking the LIMIT of the secant slope, as h approaches 0. The slope of the secant line becomes the slope of the tangent line. Not only is this limit equal to the slope of the tangent line, it is the definition of the derivative (when the limit exists) or f'(x).
4) HOW TO FIND THE DERIVATIVE OF A FUNCTION USING (LIMIT) DEFINITION: If you have to find the derivative "using the definition of the derivative" or "by the limit process", then you can use the limit definition we just found for any f(x) equation you're given. Remember that the f(x+h) part of the formula means to replace x with (x+h) anywhere x appears in the function f(x). If you want help knowing how to find the limit at the end, you can jump to my video "How to Find Any Limit": • How to Find Any Limit ...
For an intro to the concept of limits, jump to: • Introduction to Limits...
For more of my math videos for calculus, precalculus, and algebra, check out: nancypi.com

[@RR-mp7hw]

It's great to see someone thinking through what they are explaining. The sign of a natural teacher.

[@sibbyeskie]

Nancy is by far the best at explaining these fundamental concepts. Perfect amount of repetition, visuals, analogies, and pacing. And clean. I always struggle and then remember to check if Nancy explained it and *boom*, I understand it far deeper.

[@mmichii0302]

Best explanation of derivative I've read or seen. She does such a wonderful job getting the concept across.

[@wishyjoshy]

It takes true mastery of a subject to make it as simple as you do. Super grateful! Thank you!

[@rishiraisesthebarr]

A whole lotta money

[@Crashoutgavin]

💀💀💀

[@sonasunny5423]

I wonder why she isn't making videos often. Precious things are always rare. Please make more videos nancy you really are a mathsaver.

[@codeaperture]

Saving my Calculus supplementary exam

[@chibcha46]

She's made a lot of videos. I've seen many of them.

[@kebman]

@@chibcha46 A ton of them were also deleted. The ones of her former employer / company. I think that's pretty sad.

[@jaguarwolf1]

For the first time in my life I know what derivative actually means.... Great Job👍👏

[@CosmicStargoat]

An instantaneous rate of change. It is that simple.

[@CosmicStargoat]

@@ThinknSpeak I'm not responsible for your education, son. It has been decades, but I learned this in my first semester of 15 hrs. of calculus at a University. After that, it got harder with Advanced Engineering Math and Abstract Algebra, which I didn't particularly like.

[@CosmicStargoat]

@@ThinknSpeak A message to me that begins with a snarky insult doesn't deserve a reply.

[@CosmicStargoat]

@@ThinknSpeak Not at all. I can mute you with a click of the mouse. Bye.

[@HalifaxHercules]

A derivative is the same as a slope of a non linear function. When I was a Business student at Saint Mary's University during the 1990s, I took an "Introduction to Quantitative Methods for Commerce II" course, which consisted mainly of Differential Calculus, Statistics, Sets, and Probabilities. I also had to master the four different derivative rules, Power, Chain, Product, and Quotient. However, we didn't learn about anti-derivatives or Integral Calculus, but its easy to master once you learn the fundamentals.

[@Alex-hp5nl]

my gf wants me to watch a different math youtube channel bc this one makes her jealous.

[@moritz32]

ahah im dying xD

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lmaoooooooooooo

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You don't need a gf when you have math.

[@SuperSsheikh]

haha

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Hahahahahahaha

[@robnee3313]

Your lectures simply show me how well you understand what you are talking about! It's a real gift to be able to be so clear. Thank you.

[@nicolerodrigues8951]

you're literally saving my life. SO clear. My textbook SUCKS and is SO confusing. Thanks for helping me get through my homework. You explain things clearly and are not boring. Appreciate it.

[@michaelmorrison5675]

Nancy, your math videos are good refreshers for our senior engineers, especially when dealing with complex transient problems. And several of our younger production employees are interested in engineering, and maybe pursuing engineering and your tutorials are an excellent means to motivate them. Thank you.

[@cmdrbobert9862]

That was incredibly helpful, I just had my first lectures on derivatives a couple days ago, and this part was breezed over as if it were assumed knowledge. Thank you for breaking it down.

[@thanujasuraweera477]

Thank you so much for your tutorials, it really helps me when I was taking calculus 114. I scored over 80’s for my mid term exam and your lessons are very easier to understand the way you show different methods to explain things. I really appreciate your time and efforts

[@evrik78]

We all were waiting for you to make a video about derivatives. There was an enormous hole in knowledge about the topic. Thank you for your initiative.

[@sarahmt05]

Girl, you make everything so much easier!!!! Thanks - alot

[@moeidmehri2153]

clap for him

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Clap👏👏👏

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[@zeddy1929]

simp

[@sebastianpozziazzaro3576]

Finally! The clear, half graphical, half algebraic explaination I was looking for. Thanks for the clarity and the consistency!

[@DawnshieId]

She forgot to draw two planes and make it so that one controls the other.

[@cassied9327]

Studying for my final! Great refresher, I used this same video when I was learning this concept earlier in the semester. Its fun seeing the concept again after learning so much!

[@waynethurman2144]

Had so many college teachers who mistakenly thought their purpose was to weed out instead inspiring to learn and giving students confidence toward that goal. Einstein said that the poor teacher talks the simple and makes it difficult, but that the good teacher takes the difficult and makes it simple. Nancy is the best at making math understandable that I have ever seen; ever!

[@jagersilva2169]

the way you teach is astoundingly clear i have come to see math in a new way thank you!!!!

[@vocebilinguecomnathaliahar4828]

You are awesome! I have been trying to understand this for at least 2 weeks now. My teacher doesn't explain things as clearly as you do. Keep up the good work!

[@user-zb5nx9lt8i]

You're so amazing please keep posting videos back again, your explanations are an absolute masterpiece, I have never seen someone explains calculus that perfect !

[@davemusic_96]

Absolutely amazing I can’t thank you enough for making this video, I’ve been struggling to wrap my head round the definition of the derivative. I’ve watched various videos and this is the only one I’ve been able to follow! Thank you thank you thank you!!! Please keep making more videos!

[@isaiahlopez51]

This is the best "low down" I have come across! your awesome Nancy! thanks for doing these!

[@davida6146]

This video was recommended to me.... and although I'm no longer in my calculus II course, I become a bit nostalgic of the times I would stay up late watching your videos to solve just a few questions... ah good times! Thanks for your help during many countless efforts at my homework and exam preps!

[@sasbekverdi1535]

I love your videos. You're able to explain things so effectively. I swear, my professor wasn't able to convey as much info to me throughout an entire semester of precalculus and he earned three math degrees in grad school.

[@johnalley8397]

I love how personal this is. I love the way you move the secant line with your hand (Should have attached the dot to the line and moved them together, maybe). The acceleration noise is perfect. Simple, novel, brilliant; Subscribed. Keep up the great work. Really. Keep it up.

[@MrPumaKatz]

Studying the mathematics section for my FE Civil exam. Thank you for this great explanation, best one I’ve ever seen!

[@volken54]

Perhaps it depends on having just watched other videos about this topic so, little by little, I had just assimilated the concepts but, apart from this, your explanation is the clearest ever ! Thank you again!

[@HL-dw4dl]

I am from India. Thanks a lot for teaching calculus in a simple manner. All Indian teachers on youtube limit their content to stuff that is asked in exams rather than teaching for knowledge.

[@arghyasarkar6147]

Very true I know as i belong from India

[@arnavpandit6478]

sem here bhayyyyyyyy achhi dikhti h na bhay vo ??? isliye tune video khola na????

[@bilaltaj1725]

@@arnavpandit6478o bhai ghlt baat hai

[@scinc3579]

Great explanation!!! I’ve learned more with your video than I would an entire semester, thank you for that! Plus, that “interactive” writing board makes everything even easier to get a grasp of all the concepts.

[@domantassakalys1710]

She explains everything so fluently and in a gorgeous way, uses some kind of magic with her board, she's definitely a witch.

[@fahq6703]

I like how you made the line change with your hand movements. Also ur awesome and very helpful ty so much!!!!

[@rohitnautiyal7090]

I like how you explain everything with a calm and soft voice.

[@revazzhvania3795]

Very few people explain concepts in an understandable way. I mean, when you ACTUALLY UNDERSTAND what it is and why it is there. It's easy to just DO IT because you saw 100 other examples but, what it really is, is what matter the most. This was amazing.

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[@kabagaida9744]

Thanks, this helped me get a grasp on derivatives and their purpose since I wasn't really sure on that

[@jesuisravi]

I appreciate the fact that you speed up the process of writing out the definition etc....

[@tirtheshjadhav1898]

After watching three videos on the introduction of derivative, this one was by far the best! you explained this very well.

[@JakobPapirov]

After watching a number of your videos I believe that this is probably one the best ones! Having said that, all of them are great!

[@alexisramires4883]

You're awesome. Thank you so much--what my professor tried to teach in a week you managed to teach in 14:29 minutes.

[@georgevladimirovich7190]

Thank you Nancy. I am taking your videos as a reference each time when doing math with my daughters and they have both became good at it. The school marks are also confirming that. I appreciate your work and your commitment. Thank you.

[@KyIn30034]

I want to thank you. Your videos helped me passed my college math courses with either As or Bs. You explained the concepts in a much more clear way than many of my professors. Thank you b

[@VitalyBokser]

Really great explanation of finding a derivative from first principles. Always good to reiterate how its actually done without using the derivative rules. Thanks!

[@dannysalcido4738]

Let's not kid ourselves. She's gorgeous and smart.

[@38Fanda]

your mom is gorgeous and smart

[@dannysalcido4738]

@@38Fanda So is your sister

[@38Fanda]

@@dannysalcido4738 >:( youve got a big mouth

[@vargo0515]

Thats WHAT I NEED TO DO MATH!😊😍❤

[@ace6285]

No kidding.

[@sabiachauhan]

omg so helpful, and the effects are just awesome! keep putting videos

[@iangibb5297]

Best description i have ever heard, perfect session, thank you miss.

[@bluewolf314]

What a great teacher! I had a submarine captain from WWII for Algebra, and for more advanced math at a prep school, old books by a retired math old bat. I learned everything on Math by buying books and learning it myself. I have a ton of Calculus books of all sorts, but something I learn more from videos such as this. Also the instructor is very pleasing to the eye, and she certainly can draw the equations very quickly. (yes, I know).

[@simaphore]

This was the best explanation I've found for derivatives. Thank you!

[@DawnshieId]

Better: The derivative is how fast something is moving or changing. The integral is that thing that is moving or changing. The remote control is the derivative, and the plane is the integral. The steering wheel is the derivative, and the car is the integral. How fast you're walking is the derivative, and how far you've gone is the integral. There can be derivatives of derivatives or integrals of integrals, but that can be in another explanation. It really doesn't need to be a 14-minute video.

[@WomdeKynthos]

I love Nancy Pi, I have an A in calculus and I understand it, thank you miss lady!

[@erekadivine2553]

God bless this lady. I’m so glad I tumbled upon this page my calculus teacher at the university explains things in the most complicated way possible , even if you understand you’ll end up confused. But THIS HERE IS GOLD!!thank you you soooo much❤

[@srinip]

That's great! At 65, I had completely forgotten the first principles derivation, and thank you for reminding me!

[@Madmun357]

Whoa, I like how the secant line moves with your hand. Cool editing!

[@lionking2424]

I am a EE and I am getting a MS in EE. I did a bunch of mathematics. I have to say you are doing very good. Very good explanation. Do not stop. Thank you. And before I go can you also do some Caley-Hamilton theorem , Jordan method , some Eigen values and Eigen vectors video.

[@XXX-fe3or]

Perfect essential approach for teaching. Thank you Nancy for the videos

[@vectr3209]

Hey there, I completed my engineering an year ago... and just on this random day I remembered how this channel helped me out in my first year. Thanks a lot! Lots of love, stay safe and healthy. I'll never forget you. Thank you. ❤

[@sarahnamwinga6827]

Really helpful thanks you very calm when explaining makes it easy to follow☺

[@wink181]

Geez Nancy, I'm sure you've forgotten more math than I will ever know!!! Impressive!

[@CIDteenie]

OMIGOD THANK YOU!!! ❤️❤️❤️ I'm so grateful for the clarity of your videos. This was something I was positive I'd never understand and now I get it!!!

[@cayezara8110]

I keep going back to this crystal clear illustration, explanation and definition of derivatives. So simple but deep!

[@kuldeepsen3364]

Hi dear, Contact me on WhatsApp [phone] for any kind of academic help. Questions solving, help in solving assignment etc.

[@victorcontrerasiii9449]

Great, im learning derivates and improving my english listening

[@albiss1164]

I wish you a beautiful career and life! You seem to have all variables under control.😃

[@hugofiga]

wonderfully executed. everything: rhythm, clarity, visuals. well done.

[@vladimirvladimir3298]

I learnt all of this material long ago in engineering school but I so enjoy seeing it again. Math is beautiful and so are you.

[@RexonNate2031]

These effects are actually really cool, I wasn’t expecting this lol

[@Ernesto1317]

I like that feature of the crystal board and the sounds when she is writing. Good ideas..

[@drziggyabdelmalak1439]

Wow...once again you have made complex, confusing stuff very easy to unserdtand with your simple approach and relaxed everyday style. Thank you.

[@stinkyseastar3560]

Thank you so much for simplifying this concept, you're amazing!!

[@cswalker21]

I took calculus years ago. If my teacher had been this good at explaining it, I would have done much better. Now I actually do watch calculus videos just for fun. This was a great reminder of where the derivative rules come from.

[@sillabaez]

Thank you Nancy. Literally, I've been to tutoring, watched Chegg videos, other KZbin math videos, but yours is by far the most explanatory and I really get you. I like the fact that you dont try to make it hard and confusing as well as being open to new methods and easier ways for different learners. You amazing and beautiful too :)

[@melwinters9655]

Thanks. This really helped me out. Looking forward to watching more of your videos.

[@newguy6935]

Very well explained. Appreciate it. It's been a while since I've studied all this and I've been interested in brushing up on it. This really, really helps. Thanks again.

[@rosebud7951]

A most beautiful, illustrative definition of a derived tangent from a secant line. It is etched in my mind for eternity! THANK YOU.

[@abdallahigrainatt6865]

Thank you for the video😘. I follow you from Morocco and I like your videos. I am very happy and wish you a successful journey✌✌

[@tgut03]

Thanks Nancy! I learned that the derivative f'(x) is the change in the output (y) divided by the change in the input (x) as the input approaches zero. Math becomes more enjoyable once you understand the underlying concepts so thank you for sharing this joy with me!

[@KP-el7wk]

i’m in love with the way u teach. so calmly unlike my other teachers

[@badlandskid]

Oh man. This brings back memories of calc class in college.

[@keitholeniacz7602]

Not even in school but always enjoy your videos

[@brendanjames8818]

I appreciate that you can keep it simple and give comparisons to the content. Particularly when the comparison isn't true but helps grasp concepts better. I've been watching all your videos on limits, you're literally saving me in calc 1 this semester. Thank you

[@brendanjames8818]

Esp in the infinite limit videos lol

[@gabe5225]

@@brendanjames8818 Simp

[@brendanjames8818]

@@gabe5225 zZzZ

[@JesseMoshe]

This is extremely helpful and accessible. Thank you for your work!

[@donbouquet403]

Excellent explanation! I wish you had been my high school calculus teacher!

[@daxbattaglia2613]

Obviously!

[@elijahmiles3500]

My high school calculus teacher was like a 7 this girls like an 8.5

[@ferhansyed]

Simply brilliant! A teacher who knows how to communicate with a clear understanding of the kind of confusion that exists about mathematics in the minds of her student. Bravo to you Nancy.

[@Edward-yf3hp]

Great Calc 1 vids, hope you come back and do more, they have helped me a lot, thank you

[@chilly856]

You explained this way better than any of my math profs did. Its been 15 years since I took calculus and I knew the rules to get the first and second derivative, but I didn't know the underlying theory. Thanks.

[@oscarg7460]

Wonderfully clear. I have not done math for decades and your explanation provided clarity like never before. I will have to re-view your video. However, I do not quite understand what a limit is or its purpose. Again, thank you again for being a wonderful instructor. You validated my suspicion. I always suspected that math could be taught better.

[@ps200306]

Derivatives are all about rates of change. One thing is changing with respect to something else. Let's say the economy is growing with time. You see on the news that it grew 1.2% in the last year. But then you read that it only grew 0.2% in the last quarter. And, in fact, it shrank by 0.1% in the last month. Clearly it grows at different rates over different time periods. If I'm interested in the current trajectory I might look at what it did in the last week, or the last day, or even the last second. (I know that's kinda hard to do with the economy, but it's just an example). As my time interval gets shorter and shorter I get closer and closer to the _instantaneous_ rate of change, otherwise known as the derivative. You might argue that there's no such thing as an instantaneous rate, but I will argue back that I can get arbitrarily close. That's where limits come in. They let us talk about arbitrarily small things (or large things -- but for differential calculus it's always small things). Our notation for an arbitrarily small -- or infinitesimal -- quantity of time is dt. If E is the size of the economy then its derivative, or instantaneous rate of change with time, is dE/dt. It is "the limit" of the rate of change in the size of the economy as the time interval we are talking about gets arbitrarily close to zero. What use is this, other than a fancy notation? Well, the world is about nothing if not change, and derivatives crop up literally everywhere in the physical sciences as well as economics and other social sciences. Using the technique that Nancy demo'ed, plus some handy shortcuts, we can deduce a lot of useful stuff. (P.S. I went back to school after decades away from maths too, and got an advanced degree in astrophysics. Never too late).

[@batuyilmaz1464]

I already know what derivative is but I'm still gonna watch it, who cares! Right?

[@egecinel9964]

Batu Yilmaz horne!

[@axlrose5082]

@Julita Aishah Nah, is good to review this kind of things. And the tutor explains really well

[@axlrose5082]

@Julita Aishah I mean, if you see that a math video has more than 300k views and 90% of the thumbs are thumbs up, you can deduce the tutor explains things well. I would still watch it if I'm interested in the subject(like in this case). Sorry but you've lost this one, you girl

[@dineshpandey3096]

thanks ma'am, your all videos are really helpful to me & the way you speak it's great to hear.

[@mohammadorangian9843]

Thanks for the course so transparent and simply delivered.

[@SanjaySanjay-jf3gd]

You are the example of beauty with brain

[@aparnadas811]

I Literally Hate Maths But I love Watching Your Videos And Seriously I Understand Every Concept

[@rainerfs4563]

thank you for this video. I used to review as I don't touch this topic for years. Very well explained. You speak clearly and slowly and has the teaching skill

[@user-vm6ug4dj7e]

thank you nancy, your explanation is too good

[@SC-fk9nc]

So enjoyable to learn maths from a smart woman on youtube. I must nevertheless confess that it takes all my will power to focus on the explanations and not let my mind be distracted by the charm and sweetness of the teacher. :0) Good job Nancy!

[@pengpleb9523]

7:48-7:58 what kind of witchcraft is this?

[@loneventhorizon]

hahaha i was like what the fuck when those words disappeared

[@patelpruthvish1905]

wingarium leviosa… XD

[@dubiousend9429]

@@patelpruthvish1905 its leviOsa not leviosA

[@user-eh1dp1sh1z]

🌹🌹Thanks

[@nevillelusimba1689]

the math witch thingie

[@robinpettit7827]

I used the definition of limits to introduce derivatives initially and this made sense to me as well when I first learned calculus over 40 years ago.

[@tungtruong5904]

I'm learning the algorithm course and my Prof teaches some math basic. I almost forget everything from high school, thanks Nancy, your channel is so so useful to me.

[@jonathanduran3442]

I am currently studying machine learning however I never went to college so my primary source of study is basically KZbin lol. Anywho, I found this to be extremely helpful in understanding the back propagation algorithms in a neural network! I look forward to more of your videos as my learning progresses! :)

[@anishkasingh5701]

Nancy, I have my maths exam tomorrow and no matter how hard i tried I simply *could not understand* how to "FIND THE DERIVATIVE USING THE (LIMIT) DEFINITION of the derivative", however i watched this video once, and BOOM i understand it completely now, thank you SO much, you really are a life saver

[@OwnerOfOwn]

how'd it go lmfao

[@mr0o]

update?

[@namansingh5276]

1yr have passed

[@anishkasingh5701]

@@namansingh5276 haan bhai sorry 🙏 all i remember is iska concept bahut strong hogaya tha and iss topic ke saare questions answer kar paayi thi

[@namansingh5276]

@@anishkasingh5701 chlo Acha hi hua tha🙃

[@coffeeortea8428]

Extremely well explained. For the first time in my life, I am clear on the idea.

[@dafpnp]

Hey Nancy, the last time I studied all this was 25 yrs ago. I didn't know this will come back to bite me after this long. I would have studied harder and inscribed in my mind using some practicals, but it had to be this way :). Your videos are a great great help. Can't thank you enough.

[@georgelander5922]

I was wondering how they made super rollercoasters safe, now I see the applications. Thank you.

[@susankivuva5755]

I love how she teaches, really good

[@paulneilson6117]

She is not writing backwards, the image is reversed.

[@Goku-bh1ej]

Paul Neilson oh really Sherlock

[@itachi6336]

@@Goku-bh1ej no sherlock

[@Flutoid]

What is she writing on?

[@SufferDYT]

@@Flutoid glass?

[@tapuout101]

lol I was wondering if anyone else was wondering that.

[@jeffreyhamer3107]

I especially like the new technical term I just learned to decsribe a rough estimate, "wonky." Nancy makes technical stuff friendly. Well Done!

[@markw208]

Everyone should watch this video. You did an exceptional job of explaining the definition and the thinking 🤔 behind it. My Calculus Prof in college gave a similar but more basic explanation. This is a complete explanation.