# How to Do Implicit Differentiation (NancyPi)

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Published on September 15, 2018 by

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.

Follow Nancy on Instagram: https://instagram.com/nancypi
Twitter: https://twitter.com/nancypi

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: http://nancypi.com

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### 481 Comments

• Thank you!

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Thank you Nancy

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• Pliz help me this problem.
Let f(x,y)=c be an implicit function, show that
dy/dx=-fx/fy

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(in buzz lightyear voice) ‘you don’t wanna be in the way when my y detector goes off… y’

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• I need help with this one problem that I have to solve with implicit differentiation. We have to find dy/dx and d^2y/dx^2. The problem is xy-x+y=2. For the first derivative I got (y-1)/(x+1) but for the second derivative I am completely stumped and its most likely due to the algebra. :/

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• How do you know Y is dependent on X?

• Because we’re assuming X and Y here are defined in the common sense, where Y= some function of X (like when you plot on the XY Cartesian Coordinate Plane). It’s just that for implicit differentiation problems, you tend to not be able to explicitly write Y in terms of X, but the equation still gives us a relationship between the 2. 🙂

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• you say not taking derivative of y so you keep the y in 9:10. But you take the derivative in the y in 9:35. Actually why ?
I am in the mess

• In the first one she is taking the derivative of x^2y according to the product rule(x.y = y’x+x’y), in the second she has to cause you have to take derivatives of all functions on both sides

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• Can you do a video for nth derivative?

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

• 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.

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

• On the second example. Can you add dy/dx + dy/dx to get 2dy/dx. then you divide the two? but it would make the bottom equal -2x^2. So what’s wrong with my logic? Please answer

• does anyonw else get distracted at the fact she is writting mirrored. or am i crazy. like shouldnt the numbers be backwards to the camera. is she using a program to edit it to show correctly or is she actually writting mirrored.

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Parallel lines, for example. They have so much in common, but they’ll never meet!

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• It seems unsatisfying to me to have a y in the final answer. I would solve for y in terms of x in the original equation and then substitute it into the equation for dy/dx.

• Wonderfully explained, Nancy. Thank you. I still need to view many of your earlier videos in order to establish the basics. Is there a particular order to your videos, going back to Pre-Calc ? If so, I would be most grateful to you if they could be listed and presented as such. If not, I understand perfectly. I’ll go back and research them. I am a subscriber and was also a subscriber when you were Bffmath. Many thanks to you, Nancy. And thanks for explaining techniques with such precision and patience. You’re amazing.

• Thanks so much for helping 😀

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• wait are you writing backwards???

• But Y, so serious 😛 hahaha

• How do you answer ×= sec 1/y ??

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Thank you for doing these.

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• Hey Nancy, could you do a video touching on related rates? Thanks!

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• On problem #2 why did you apply the product rule like that?
I thought product rule is
f(x)g(x)=f'(x)g(x)+f(x)g'(x) so it should be -(2xy + x^2).

• John Smith because addition is commutative……..?

• “Y detector” 😂😂😂…anyways thanks…simple enough

• Thank you so much. your second example really touched up on other areas like product rule and algebra. Does iD involve things like quotient rule and log or natural logs. I’ll try to find out that answer before my upcoming calc test lol, if you or someone else doesn’t get to it before.

• 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.

• Help me please
Let f(x,y)=c be an implicit function, show that
dy/dx=-fx/fy

• Nancy Pi you are amazing, thank you!

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• THANK

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• The problems that kick my ass are like….

If xy+5e^y=5e , find the value of y” at the point where x=0

And

Finding dx/dy of:
cos(xy)=1+sin(y)

• second one,

cos(xy)=1+siny
xy=arccos(1+siny)
x dy/dx + y= -1/(1-(1+siny)^2)^1/2 * cosy dy/dx
dy/dx ( x + cosy/(1-(1+siny)^2)^1/2 )=-y
dy/dx= -y/ ( x + cosy/(1-(1+siny)^2)^1/2 )
Where did you get such questions xdd

• and finally
xe^y=x-y
im guessing you need to find dy/dx?
y=x-xe^y
dy/dx= 1-(xe^y dy/dx + e^y)
dy/dx=1-xe^y dy/dx -e^y
dy/dx(1+xe^y)=1-e^y
dy/dx=(1-e^y)/(1+xe^y)
there you go

• @Warc Arden I got them from webassign.net

• @Warc Arden thank you so much for your time! I have a final coming up, so these will be very helpful🙂🙂

• nvm the second one is wrong, I thought you had to find dy/dx
let me fix that one for ya,

cos(xy)=1+siny
xy=arccos(1+siny)

differentiating with respect to y,
x+y dx/dy = -1/(1-(1+siny)^2)^1/2 * cosy
so
dx/dy= -1/y(cosy/((1-(1+siny)^2)1/2) + x)
there ya go

(Yeah its just the reciprocal of my original answer but just wanted to do it again with respect to y since thats what is meant to be done)

• 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?

• I Ned that too also optimization

• Same

• 1:42 I thought that sound was my stomach.

• thanks

• Nancy pls do a video on related rates ;-;;;;;;;;

• wonderful teacher, thank you!

• Your voice is just relaxing

• I guess there is a little mistake in your summary. Instead of “…y^2 times dy/dx”, it should be written: “…2y times dy/dx”.

• Wow she’s really good at writing mirrored and backwards wtf
Unless they just flipped it in post

• Thanks mam

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• Diana get ready to get F

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• Wow all you guys are such assholes

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

• Yes

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• violet from Incredibles??

• Thank you so much…It was so helpful!😊

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• holy shit i finally understand

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• ily

• Shouldn’t there be a dx after the 2x when differentiating x^2+y^2=36? Meaning we get 2x dx + 2y dy = 0

• No, we are differentiating with respect to x here.
thus derivative of x^2 with respect to x is simply 2x.
derivative of y^2 with respect to x is the derivative with respect to y first and as per chain rule, then with respect to x which is why we write it as 2y dy/dx

• Your videos are amazing. keep it up.

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

• You are a gift

• 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!

• It’s very easy tour trick. You write as always then mirror the whole video. That is why appears to be you are goofy

• THANK YOU!!!!!!!!!

• Excellent your teaching method is very good thank you.

• THANK YOU

• like

• at 12:46, Nancy’s marker magically goes from a pink marker to a black one…

• omg thank you ❤️

• Thanks for your help.

• We are using chain in every question of differentiation ex. d(X^2)/dX = 2XdX/dX

• Thanks for video

• Does anyone hear a faint siren sound in the background ? Or is it just me ?

• I have a graph with me. Which have only phase of 180 degrees. But have a length of 1000mm. And a maximum height at 90 degree of 1mm. How can I find out the equation.??

• I’ve never been able to work out how she is able to write backwards.

• EightLeafClover she doesn’t

• @Huda Mursal What happens then? How does she manage that?

• She mirrored the video

• U r soooo hot

• Great video. Thank you.

• Can’t we convert the equation into y=√(36-x^2) and then explicitly differentiate it?

• try to do it.. and compare the results 🙂

• derivative of y=x^2 with respect to ln x

• @Abhinav Jain thank you for your reply.

• Why were you so shy while recording the video by the way thank you very much

• Thank you so much Nancy !

• 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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• useful – thanks

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• Hi

• How to simple power defferentiation ?????

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• I just love the editing like how you use a mirror for teaching purpose 😍

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• x4y2+sin(y)2=0 plz solve this on ur videos “4,2,2” r powers

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God bless u

• You only think you’re impressed but then you notice she’s doing it all while writing backwards.

• 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!

• An excellent teacher.

• her face flushes around 10 minute mark in these derivative series. I don’t know if it’s stress or something else lol

• Best help for my diff equation

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A Teacher should be an actor like her.

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• Total and partial differentiation please

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It would be much appreciated since I just started at university. Thanks

• i love u, thank u

• Lose the pink marker. It is not showing up well on your board. Stick with black, blue and green markers!

• How are you able to write on the glass board from the rear?…good work

• Ramy G i’m kinda thinking that she writes it facing her and then they just flip the video so that it faces us now

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• implicit differentiation? pSSHSHSHSH!!

• haha

• This is more correct
Y^2+x^2=36
y = (36_x^2) under root
Derivative of( 36_x^2) under root is
(_2×)/2×root of (36_x^2)
_×/root(36_x^2)

• You are awsome!

• My eyes only focus on her not on her notes. HAHAHAHA

• how tf do u write backwards

• You’re indeed a true hero. Please have more and more example solvings. Thanks very much, Nancy!

• Thanks nancy!

• What about if it is written with respect to y? Means all that has x term will be multiplied by dx/dy?? Need help please

• could you please do a video on how to find points of intersection between a tangent and a curve, im so lost…..

• Hey Nancy what does 2x rhyme with 😘.

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• Thank you!!! Finally I understand implicit differentiation!

• You actually can solve it without implicit diferentiation, look:

First we need to isolate Y:

x² + y² – 36 = 0
x² – 36 = -y²
sqrt(-x²+36) = y

Then do the derivation normally:

dy/dx = 1/2*sqrt(-x²+36) * -2x
dy/dx = -2x/2*sqrt(-x²+36)

Since sqrt(-x²+36) = y, we have:

dy/dx = -2x/2y
dy/dx = -x/y

• Implicit Differentiation is a short cut for solving multiple intermixed variables. Your way is fine for non-intermixed terms, and longer. Besides, there are more than 1 way to solve certain MATH problems.

• Great way

• Not only is she writing left handed but she writing backwards too

• at 7:16 i got a little spooked

• thankyou

• Thank you so much. 😍 Your so Genius and Sooooo beautiful. 😍😍😍

• 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!

• God Bless You!!!! My calc teacher isHORRIBLE and this saved me

• am I THE ONLY ONE who can’t see the pink written stuff?!

• This video helped but would have been better if you did a problem that had a x^2y^2 in it

• Hey Nancy pi how do I find the second derivative using the implicit differentiation?

• Here’s the start of my question: xy+y^2=1; solve for y”

• Sister pls do some videos on complex numbers and permutations and combinations. Your way of teaching is very good and understandable thanks for it.

• Thanks the test is tomorrow and you just helped a great deal.

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

• Nancy you r beautiful and thank you. I want to know about vector

• Thank you so much Nancy you are making things a lot easier for me now. BTW you are so beautiful as well, I almost lost concentration by starring at you hehe. Watching all the way from the land down under (Papua New Guinea).

• ❤❤❤❤❤❤❤❤❤❤❤

• frustrates me that it appears as if she’s left-handed but righting backward right-to-left for the viewer. could so much skill really be possible? in need of a behind-the-scenes.

• Christian Evans 99% sure that what she does is write on a transparent board from behind it just like anyone would and then she simply mirrors it when editing the video so that the writing is right

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

• very beautiful woman with brain thanks a lot

• Thank you so much!!

• Wonderful Teaching🤩

• I don’t know if this is a silly question…Are you writing in reverse? or on a mirror? I’m getting distracted with trying to figure out how you’re writing and not paying attention to the content.

• Your videos are just basics
So kind of wastage of time, try covering some concepts for JEE exam

• Great video 👍

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

• thanks nancy

• 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.

• watching your fantastic videos from Ethiopia

• Thanks madam

• Warning! 13:29 “this looks really innocent.”

• Plz tell me what is derivative of y

• Are you writing backwards?

• Hi,thanx,thee be incredible,now GOD BE BELOVING OF THEE,THY TRUTH BREATHES LIFE INTO LIFELOVE OF THE RELARTIVE TRUTH OF SOME SYSTEMS,SEE 3D,GRID IN LARGER 3D GRID,DFRNT AXES MEETING POINPARRICLE,THE PAIR 3D GRIDS,THART THE d/D,DIFUSN/DENSITHRUM OR FISDION/FUSION,OF EACH 3D GRID,WITH DFRNT RELARTIVE dt,d^2t,of 2 grids,as well as cenchienrum,functions dascribing trajectory of each axes meeting POINPARRICLE,both breathes in rhythm,or dissnanc,inflection exhaustion,into dot,dt=>,thus die,death of system,Pi best seen as dmtr of thick theory straight line rotarting 185-190,370 degree to ensure circle true,if plane dmtr rotartion flart,RELARTIVE smooth,but theoretical,as 3 dfrnt RELARTIVE,3 x 3D grids see in 11d,to 12d,but see axes as each dfrnt rartRARTIO RELARTIVE MAESURE system reliabilithrum,as tuning drum alters with thin,or thick artmaosphere,thick,low pitch,similar with thy functional wissum,breathing truth,Pi varium,inside,ARROOUNDING,scribe circle ⭕️ dmtr longest str line scribing circl,thanx,bliss, XAMBUNJILOM

• Thank you

• 9:09 NO. WAAAAAY.

SO THATS why we don’t mess with the second y…. omg. *kaboom*

• i cant tell if ur sarcastic but I don’t get this part at all help lol

• @Yousef , she’s using the product rule to differentiate x²y. Remember the product rule is f'(g(x)h(x)) = g(x)h'(x) + g'(x)h(x). In other words, the first function times the derivative of the second, PLUS the derivative of the first function times the second function.

So, with x²y the first function is x² and the second is y. (Remember that y is just shorthand for the function y(x)). So we have the first function times the derivative of the second: x²(dy/dx). PLUS the derivative of the first function times the second function: 2xy (note: no differentiation of the second function here).

In total we have: (d/dx)x²y = x²(dy/dx) + 2xy.

• Example:
Implicit differentiation of 3x^3 + 5y^2 = 15:

3x^3 + 5y^2 = 15
(The differentiation is implied)

• You’re a really, really great teacher. Very clear

• @7:18 she stares into your soul…

• 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!

• The whole video i just watching that she writing in left hand in mirror form….😑
And enable to understand any implecist. Stuff🙁

• 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.

• Thank you. Helped me on the final

• 4:07 – Why do we “have to” use implicit differentiation here? Why can’t we get it inti the form “y = …” by subtracting x^2 from both sides, taking the square root of both sides (hmm, how would the +- be handled during the differentiation? We would possibly get two results, right?), then differentiate normally?

• ~ 8:40 – The derivative of y with respect to y is one, but with respect to x it would be 0 (if it didn’t depend on x, that is).

• كيف تكتبين كذا اتمنى الرد 💙

• عبدالله ما أتوقع انها تعرف تقرأ عربي، بس أكيد استخدمت لوحة شفافة للكتابة.

• Can you teach me long division. Mamm

• Nicely done! How about doing a video on Fourier Analysis?

• Or bezier partial differential equations ? Thanks!

• Thank you nancy. For your help in learning the derivatives

• Nice explanation. Love you

• How can you not e understand Math, when a sexy & intelligent woman like Nancy is explaining it to you?

• I wish I had this available when I was in college.

• sugar britches i want you to balance my check book

• you’re not a math teacher, you’re a model

• 4:11 You can in fact do it explicitly, its just easier to do it implicitly. Anyways good video!

• do you edit your own video?

• Stunning smile!

• She sounds so bored. “So this is X negative = kill me now”. Explain how you’re more boring than a pure mathematics text book?

• You should do a Netflix series

• Beauty with brain

• ❤️❤️❤️❤️🌹🌹🌹

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

• Liked, very understandable

• So difficult to concentrate becoz she is too good looking

• Nancy, are you doing this in a mirror?

• Miss NancyPi writes backward with the left hand, amazing

• She fine

• very beautiful and intelligent. if i proposed, will you accept me

• THANK YOU!!!

• May God continue to bless you

• With me discovering this channel, I’m gonna pattern calculus easy. Mans gonna secure that B grade

• that B = BAG

• She speaks energylessly

• They say beautiful girls don’t understand math. They must have missed this Nancy

• Hi Nancy!

• how can you right by left hand and it is a mirror reflection, omg good practice

• omgosh I’m cramming for my Calculus midterm. Why didn’t I find your channel earlier in the day! <3 thank you!

• y^2 + x^2 = 36, how is the answer -x/y help? Interesting

• When you have ” x^2 + y^2 = 36 ” why can’t you simplify this expression to “y = x+6 ” and “y = – x -6 “? Like can’t we just simplify and then differentiate?

• Can we use y’ instead of dy/dx?

• just notice, how is she writing the opposite way lol?
but what an explanation
thank you so much

• Thank you so much!! I feel like I’ve made every mistake, but I get it now, thank you for such a clear and helpful video!

• hey Nancy I love you and I have some questions, for question 1 why not just put it in the form of y as y=(36-x^2)^1/2 and is your last name really pi or you just want to be irrational 😉

• WHY AM I WATCHING THESE MF ADS IF ITS NOT YOUR JOB

• THANK YOU

• thx woman

• Your can write backwards cause your left handed. That’s the trick

• Its so rare to see beaury with brains

• Corona Virus screwed my cal1 class and this fr did help

• Thank you very much!

• Now I’ll just watch it again so I actually understand what she’s saying…

• thanks so much!

• I really feel like we could use a blooper/outtake video from you.

• How the heck do you write backwards so neatly

• I enjoy your videos and all but I wish you would write the dy/dx in a darker color than pink like purple or something else that will be easy to see.

• Thank you

• Please diffrenceate both side
this
X^2+y^2=k^2

• Help!

How do you solve

y=x^2 / y^3

• Beauty miss

• why do you keep sighing like your being held at gunpoint to teach

• Why can’t you rearrange a function like x²+y²=36 as y=sqrt(36-x²) and then use the chain rule by writing that as y=(36-x²)^0.5?
Also I liked your video, it did help me.

• Okay I tried it and it works, I guess it’s the same thing

• Wooohoooo today is my last day of Calculus ever and I believe I will finish with an A :)))) thank you for your videos! Most math videos are so boring!

• u got me thru ap calc queen

• Who is watching this video in 2020

• How can you write in reverse that’s AMAZING

• So, she’s writing from right to left. I really like her commitment

• It’s just flipped, lmao

• @s!CKman k

• You’re so hot and sexy

• Can you please share a video on how to solve this
Xsquare + Y square ÷ X square – Y square = a ( a is constant)

• Great video.

• I’m going to always remember ‘it should set off your y detector!’ LOL funny but thanks for the reminding tool haha <3

• Thank’s a lot, it helps better than the one with my mothertounge tho

• Are you writing everything in reverse on a glass plain that you’re filming through?

• Good

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

• its mirrored after

• @Google User I think he meant that she’s writing away from her body

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

• @Google User She’s writing right to left ..

• @Trevor Starr yes but if you mirror the image it becomes right to left

• I need help to find the general expression of X belongs to the vector field manifold which satisfies : [ d/dx + d/dy , X]=X

• You could also use Partial Differentiation as a short cut for Implicit Differentiation

• I’m not looking at what you are writing, just look’n at you

• Thank you! I might pass AP calc now

• Can you please make a video on intense or some of the hard problems related to integration ??

• great video

• Thank you dear Nancy 👍👍

• Thanks Nancy
You truly are a gifted mathematician
God Bless You

• Ur videos r very helpful

• I want to marry her.

• She’s beautiful tho, so now i kinda wanna study

• 5 mins and 48 seconds there’s a police siren

• When my teachers teach : Oh Calc is hard
When you teach : tbh thats pretty easy
Plus my teachers arent bad at teaching maybe i find it easy now just coz in re watching what i earlier learnt

• This makes so much sense now… Thank you Nancy!

• You’re so cute

• Nice

• You are very smart, and surprisingly nice to listen to… Thanks for this video it helped a lot helping me to start to understand what to do, also I never knew you were double jointed til I seen your pinky bend backwards at the start of this video. 😛

• Thanks Nancy!

• what i really want to know is how does help me in life? when do i use this? or have i been right all along and postsecondary education is just a scam to steal your money, break your soul and erase your humanity in preparation for the professional world?

• Right explanation

• These videos are awesome, I was so confused but this cleared it up better than the 3 hour lecture I was in! Thank you

• Nany I reallylike your vids i suscribed liked and hit the notification belll you make it look so easy

• Love your explanations, very clear and concise, making math, especially calculus, easier and fun to learn! Much appreciated

• Thanks. I wish they would hire you to translate Pearson textbooks instead of the incompetent lizard people who eat there own feces and use Google translate 30x in a row before defining a theorm or anything important…

• thank you for smiling nancy 😀

• am now a lecturer cos of you, thanks my tutor……NANCY

• I did a problem like this 23 years ago. I think it’s the same one I totally don’t know how to do it anymore. A co-worker and I had to figure out a hole size and I told him we could use a Bering to measure the hole size. He didn’t believe me at first. Them I drew a picture of the bearing sitting on top of the hole and said if we can get the height of the Bering we can get the hole size.

• He then was amazed and said oh ya I know how to do that!

• Thank you

• This was a big help thanks prof pi

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