Implicit Differentiation

[@TheOrganicChemistryTutor]

Derivatives - Free Formula Sheet: bit.ly/4dThzf1 Final Exams and Video Playlists: www.video-tutor.net/ Full-Length Videos & Worksheets: www.patreon.com/MathScienceTutor/collections Next Video: kzbin.info/www/bejne/gZO8iGx8edJgkM0

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I've spent days trying to get my brain to understand implicit differentiation. After dozens of videos from class and other youtubers, yours is the first one to have it make sense to me. Thank you so much!

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[@georgesadler7830]

MR. Organic Chemistry Tutor, thank you for an excellent video/lecture on the classical Implicit Differentiation in Calculus I/II. This is an important topic in Mathematics and Engineering. This is an error free video/lecture on KZbin TV with the Organic Chemistry Tutor.

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[@dong-wooklee]

Shortcut for implicit differentiation Take all the things on one side then do as below: dy/dx= -(diffentiate wrt x, take y as a constant)/(differentiate wrt y, take x as a constant) Eg. Lets say I need to differentiate x³+3xy=y²+1 Step1 (taking everything on one side) x³+3xy-y²-1=0 Step2 differentiate x³+3xy-y²-1 wrt x and take y as a constant That would be equal to 3x²+3y Let 3x²+3y = m Step3 differentiate x³+3xy-y²-1 wrt y and take x as a constant That would be equal to 3x-2y Let 3x -2y = n Step4 Apply the rule and obtain dy/dx dy/dx = -m/n = -3x²-3y/3x-2y This may look like this is quite lengthy but in actual it hardly takes 10secs. I wrote this long so that anyone who sees this could clearly make sense of it and understand

[@aooide]

Interesting method, thanks.

[@Maroofgilani621]

Thank you sir may I ask you a question

[@Maroofgilani621]

7:23 how did -2x=cos xy² become y²cos xy² ??? Where did y² came from ???

[@havagoodnight2724]

@@Maroofgilani621 he distributed cos(xy^2)

[@drofeng]

@@Maroofgilani621 Chain rule : d/dx sin(u) where u = xy^2 d/dx (sinu) = d/du (sin u). du/dx d/du (sin u) = cosu du/dx = y^2 + x.2y.dy/dx Then cos(xy^2) . (y^2 + x.2y.dy/dx)

[@j.o.5957]

It's so gooood, realizing that d/dx y^3 = d/dx * y^3/dy * dy = d/dx * 3y^2 * dy = 3y^2 dy/dx made my day

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[@dkke825]

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[@joekerry2206]

This gave me the key points needed to proceed with implicit differentiation.

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wow much love from uganda thanks for simpifying implicit differentiation for me

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Thank you!!! this video just makes so much more sense than the textbook and class!

[@voodoochild24262]

gah i am so confused why we can use dy/dx = -x^2/y^2 in the quotient rule result but not for an immediate implicit differentiation of 3x^2 +3y^2(dy/dx) -----> 6x+6y(dy/dx)+3y^2(d2y/dx2)(dy/dx). it obviously doesn't seem to math out to the same answer....just feels weird that this doesn't also work you're the best as always!

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[@EliVang-v6q]

13:35 I believe you could have substituted 9 in for (y^3 + x^3) as well, thank you for the help!

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[@dreadhaisler6054]

12:56 To add, notice that y³ + x³ is also equal to our original given/expression x³ + y³, and it states that it is equal to 9. So subtituting the value of that expression will yield to a same answer. This is the beauty of second derivate in implicit, original expression tends to have a relationship on some part of it.

[@nokuthulamakhunga7039]

I have a question,so you mean second derivative will always give you the answer of the original equation if you substitute???

[@dreadhaisler6054]

​​@@nokuthulamakhunga7039 Actually, there are times that a given will not give you coordinates like (1,2) etc. But always remember that in implicit differentiation - second derivative, there's a higher chance that your solution will consist of the original given equation like for example here was x³ + y³ = 9, only the majority though, because some are not. Sometimes, you have to make little algebra tricks to make show of the original expression. For example, instead of getting y³ + x³, you get -y³ - x³, what you're going to here is to multiple it by "-" or -1, so it will become -(y³+x³) which will be equal to -(9) = -9.

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8:22 , is it considered when you simplify the final answer by expanding it.?

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[@balllallalllalllallalhqbal2798]

at 8:24 , could we cross cos(xy^2) in both numerator and denominator

[@Th3Name]

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[@bigcrimsy]

The thumbnail scared me with the -x^2/y^2 i thought everything i knew about derivatives was wrong lol

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[@mattmo8648]

I was totally following until you changed from dividing to distributing at abouit 6:48. I didn't track how the distribution worked.

[@mattmo8648]

particularly when you distributed it to the derivated of xy^2. I don't see how the x ended up in the derivative of 1y^2, instead of it being 2y, or what happened to the 2y where it's x(2y(dy/dx))

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Thank you so much

[@zackgrey4472]

Does anyone understand why you add the (dy/dx) and how to know when to add it? I'm confused, how do you know when to put the extra differentiation

[@brianxchengx648]

You add it when differentiation y variables

[@jonathanjeffrymulyana4390]

You put it on every variable except x

[@jonathanjeffrymulyana4390]

Because dx/dx cancels itself out

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[@shatteredknight1129]

1:00 You don't really add the DY/DX. You times. But including it is a figure of speech. So it's a figurative add. But it will be taken literal and confuse users.

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So I don't have to write it out on paper like that?

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[@comprehendnature2404]

As I understood it now, implicit differentiation is simply derivation with extra precision to what we are differentiating in respect of what. dy/dx. dr/dt. dv/dt ...etc. But normally d/dx is d to respect of X assumed and does not follow the rest of calculation.

[@MrDarren690]

Yeah I think it attempts to relate y and x even though they’re visibly (in the equation) not in the regular y = x form. I keep hearing that it applies the chain rule, but if that were the case, then we’d need to treat y not as a single variable but as y(x) or a function of x. (Chain rule in y(x) 👉 would be y’(x) • x’.)

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Why do we take the second derivative to find the point on the last problem but not the ones prior?

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[@yishenjhs6819]

@@valawee Oh sry, completely forgot about it. I got a 38/39, and I was pretty happy with that