Partial Derivatives and the Gradient of a Function

Рет қаралды 193,130

4 жыл бұрын

We've introduced the differential operator before, during a few of our calculus lessons. But now we will be using this operator more and more over the prime symbol we are used to when describing differentiation, as from now on we will frequently be differentiating with respect to a specific variable, and we will have to keep track of which one it is. This leads us to the concept of partial derivatives. Although partial differential equations sound like extremely advanced math, and they will get pretty hairy a little later in the series, they're aren't too daunting when just going over their definitions, so let's see what they are and also learn about the gradient of a function, which involves partial derivatives.
Script by Howard Whittle
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Пікірлер: 163

@KidDroskii 3 жыл бұрын

"The gradient is kind of like a slope for a higher-dimensional function" is just what I needed to hear to conceptualize this. Thanks!

@adityaadit2004 4 жыл бұрын

Best 10 minutes and 56 seconds of my life. Such a clear explanation!

@SumriseHD 3 ай бұрын

Fuck Intercourse, watching Dave is the real deal ❤

@davidbowman9695 4 жыл бұрын

So good this explanation makes MIT lectures look like an overpriced DLC pack

@maxcl3474 3 жыл бұрын

🤣

@aashsyed1277 3 жыл бұрын

ho can proffeser dave explains heart this comment?!

@samsonlawal1600 2 жыл бұрын

Man said "DLC Pack" lmao

@saf8514 2 жыл бұрын

same with Harvard lectures lmao

@hemantsharma17 2 жыл бұрын

totally agree.. he explains algebra so nicely..i tend to sleep when i listen Glibert's lecture :-)

@onesun3023 4 жыл бұрын

Gradient: A vector made up of all the partial derivatives of the function. Thank you!

@ycombinator765 3 жыл бұрын

Wtf is Jacobian then??

@Darkev77 3 жыл бұрын

Pendulum theSimpleOne lmk when you get an answer

@tejaswinikarpe3911 3 жыл бұрын

@@ycombinator765 jacobian is something that we use while changing the cartesion coordinates to some polar coordinate system or transforming to any other system.. it gives the amount of change that occurs in area after transforming. Eg: if cartesion coordinates are converted to spherical polar coordinates we have to substitute rho^2 sin(phi)d(rho)d(phi)d(theta) for dxdydz.

@ankitaaarya 3 жыл бұрын

@@Darkev77 he got the answerrrrr seeee

@Darkev77 3 жыл бұрын

@@ankitaaarya beast

@strugglingcollegestudent Жыл бұрын

This man carried me through general chemistry, calculus II and now calculus III. I can't thank you enough for teaching me math.

@kahdargo7 4 жыл бұрын

This is awesome. Feel like I finally understand the gradient now.

@radiacia_3511 9 ай бұрын

dude, you are capable of explaining multivariable calculus to a 15 year old so that he can actually solve questions, youre a God (I'm the 15 year old)

@StephenGillie 4 жыл бұрын

Calculus has been independently created/discovered at least 4 times (Pascal has a programming language named after him for his version) and thus we have 4 completely independent, yet internally consistent, sets of notating these concepts. Why did it get reinvented? Lack of local higher mathematics textbooks and education. Thank you for helping to fill this void in the current age, and making a 5th time unnecessary.

@Terry_Hastur 9 ай бұрын

That's cool to know. Thanks. Thanks to you too Professor Dave.

@idealthinker101 2 жыл бұрын

He taught in such a visual way... ! I couldn't "understand" this concept in my 6 months course of Partial Differential Equations. So I just crammed some formulas and just passed the exams.

@jackanderson8278 3 жыл бұрын

I love how clear and concise your explanations are!

@danielcheong4804 2 жыл бұрын

this is what you call a video. One of the best teaching videos I've ever watched

@josephabboud1151 2 жыл бұрын

Thank you for all your great videos! I'm studying for finals right now and your videos are infinitely better than my professor who lectures online from his bed. You make it so easy to understand in your great verbal and visual explanations, and your videos bring a lot of fun back into learning! I'm into engineering and I love learning through your videos on biology and astronomy and anything really, because learning is fun and awesome. You're a great teacher, thanks for doing such an amazing job :)

@pulpettedilmare9597 4 жыл бұрын

You rock man! gracias amigo. Im so happy to finally understand it

@goutsugoutsu 4 жыл бұрын

Oh thank you so much! After 19 years I can finally picture it!!!

@alejandroalmarza8447 2 жыл бұрын

Profesor You are amazingly clear…like all my colleagues here say your 10:57 seconds video summarized hours of calculus classes.. my admiration to you. Thanks

@GAment_11 3 жыл бұрын

I just went down and liked every comment that was positive on this video. Its the only way to extend my appreciation! Thanks man!

@sreedhar75perupally 2 жыл бұрын

Dave Sir 👍🙏 Sir you are not just a Professor, in fact you are God sent Educator for all the students & Ex Students like me who studied Calculus 30 years ago ( 1991-1992 ) during my intermediate College days (11th Class ). From past two months i have been watching & already watched more than 40 Calculus lessons on your KZfaq Channel. Sir i Thank you & Salute You.

@te-kowski Жыл бұрын

Literally the best explanation. Trying to do a project where partial derivatives come up, and I needed a quick refresher.

@devanshujoshi8393 3 жыл бұрын

This is highly underrated stufffff Ngl I’m lucky i found this 🔥 I subscribed❤️

@qualitytoolbox4872 4 жыл бұрын

An eye opener video. Neat and tidy.

@buraxta_ 2 жыл бұрын

the coolest and prettiest explaining teacher I've ever seen!

@gandalfthegaytwotowerdestr3391 4 жыл бұрын

What a Life saver, thanks so much, professor

@Deepak-pi9xx 3 жыл бұрын

Thank you so much. Finally understood the real meaning of partial derivative and gradient. 😇

@MohammadBenSalamah 4 жыл бұрын

Excellent explanation!

@amanjmullick2930 4 жыл бұрын

Do you teach all subjects?👍good work btw....

@MinhLe-xk5rm 4 жыл бұрын

wow, amazing video. please keep making more ML videos!

@theologyscienceandpropheti6808 4 жыл бұрын

Thank you.... happy teacher's day

@kaneezfatima926 3 жыл бұрын

Wow You have explained very good Finally I understand this concept Keep it up.

@Borntowin894 4 жыл бұрын

Was the video time 11 mins😲I didn't have a feeling that 11 mins have passed by. it was deeply interesting.thanks sir🤗

@tharunraj9974 4 жыл бұрын

God!!!!! You saved me !!!!! I have test tomorrow on this topic !!!!

@rigbyb 2 ай бұрын

Really helpful video, thanks so much :)

@Dennis4Videos 3 жыл бұрын

Clear as water, helps me understanding Deep Learning!

@elenaroyss7810 5 ай бұрын

Thank you very much! It is the best explanation of partial derivatives that I ever heard!

@elharithhashim4424 3 ай бұрын

Very clear explanation thanks

@omer7895 2 жыл бұрын

How would you find the gradient of f(x(s),y) is it still d/dx, d/dy or will the chain rule need to be applied?

@HeathWatts 8 ай бұрын

Nice review of gradients! Thanks!

@sureshtanwar3588 4 жыл бұрын

happy teacher's day sir.....

@FD-rt3rv Жыл бұрын

Fantastic explanation

@shlokekhullar4261 2 жыл бұрын

Thankyouuu soo much professor….absolutely incredible explanation!!!!!

@portgasdace8961 4 жыл бұрын

Just awesome !!!

@88NA 11 ай бұрын

Thank you Professor Dave

@antonbreugel3332 4 жыл бұрын

Hallelujah, just saved my calculus...

@sambananas4513 4 жыл бұрын

Thanks for making that so simple for me @ 59. Cheers!

@sstein5866 4 жыл бұрын

Great explanation! Just one question: Why does the gradient point in the direction of maximum slope?

@fineartpottamus9020 3 жыл бұрын

due to the addition of the partial derivative vectors using laws of vector addition

@carmelwolf129 2 жыл бұрын

@@fineartpottamus9020 this was the final puzzle piece for me, now it all clicked together. thank you a lot.

@carultch 10 ай бұрын

Directional derivatives tell you what the slope will be, along a given direction among the input variables. Taking a sweep across all possible directions, you'll see that the maximum possible directional derivative occurs when the direction among the input variables is parallel to the gradient. To find a directional derivative, you form a unit directional vector, and take its dot product with the gradient vector. As an example, consider the function z = x^2/8 + y^2/4, at the point (1, 1). Suppose we're interested in a direction that is along the diagonal of a 3-4-5 triangle, that is roughly 37 degrees from the +x direction. Our unit directional vector (u) would therefore be given by u = . The gradient at this point is . So the dot product gives us 0.4 + 0.24 = 0.624. This is the directional derivative of this particular function. The maximum possible directional derivative at this point, will have the same direction as the gradient. Its unit vector will be . Taking the dot product with the gradient, and we get 0.75/sqrt(5) = 0.335. This is the maximum possible rate of ascent.

@con_el_maestro3544 11 ай бұрын

I watch this channel so much that I once had a dream and your theme song made a cameo 😂

@pusheletsommatladi4686 4 жыл бұрын

Okay the content of this video is super but the Intro always have me like 😂😎

@nancysanskriti2158 3 жыл бұрын

Just got to see ur videos sr..... U are an super osm educator... Lots of love ❤😘

@elizabethsimakando7299 3 жыл бұрын

This is very helpful. Why don't you do a video on higher order partial derivatives and total differention

@simantajenaadvancedmathema9764 4 жыл бұрын

Good explanation sir

@duyanhtran4723 7 ай бұрын

Thank you.

@ahmedelsabagh6990 3 жыл бұрын

Great teacher

@giorgosrallis7044 3 жыл бұрын

Great video

@naders. 2 жыл бұрын

Thank you! 😊

@ryannkohlman5751 3 жыл бұрын

Wow great explanation. Sucks to see us students pay for an education where profs have a hard time explaining clearly. Thank You!!

@vikramnagarjuna3549 4 жыл бұрын

Thanks sir, clarified. Please do on line integrals and Greens theorem..

@ProfessorDaveExplains 4 жыл бұрын

those are coming!

@mrhatman675 3 жыл бұрын

Omg now that I know what it s definition and what it means I can work out what these beatifull weird equations mean thank you!!!!!!!!

@sotiris41664 Жыл бұрын

Even a 14 years old student would understand the gradient of a function with this video. I am not kidding I am 14 and I finally (after 5 days of search in internet) understood what gradient is.

@kaan7120 2 жыл бұрын

thank you so much you are the best

@ayushagarwalroll0283 3 жыл бұрын

thank you sir.!!!!!!

@jamespatrick9191 2 жыл бұрын

Hi! just a trivial question, what does "In" in "2xIn(y)" stand for?

@kashyaptandel5212 Жыл бұрын

natural log, it’s logarithm of y with base e, (or which power would you raise e to , to achieve y)

@aniketjoshi1610 4 жыл бұрын

Thank you sir! I wish you to make vedio on Total differentiation. Please ! Please! Please! Please!!!!!!!!!

@coolwinder 4 жыл бұрын

Yeah, I get the gradient, but I am not sure I do total differential. You can also mention gradient of an error function of a neural network, as an example.

@asaidinesh5220 4 жыл бұрын

Hope u make video on divergence and curl of a function, its goona make my visualisation much clear😁...by the way tq sir for the gradient video...😇

@ProfessorDaveExplains 4 жыл бұрын

that's the next one!

@santinacasari311 Жыл бұрын

Valeu!

@ZYau-lc5ql Жыл бұрын

Hello, why does the grad f(x,y) have the component of z-direction? I mean if the gradient of f(x,y) points in the direction of maximum change, that would be a z-direction.

@carultch 10 ай бұрын

Gradients of a function of multiple variables, are limited to the space of the input variables. The gradient of f(x, y) only exists in the x-y plane. It represents stuff that is happening in the z-direction, when f(x, y) is represented as the z-position in a 3-D spatial coordinate system, but the gradient itself doesn't exist in the z-direction.

@abhradeepghosh7102 2 жыл бұрын

The lecture is awesome. Clear and precise. But the answer to the gradient of the function at (4, 1) should be (1/2, 0) cause ln(1)=0.

@samurainair1 Ай бұрын

Awesome

@abdullahalaraz7404 2 ай бұрын

But I don't understand why x axis + y axis vector will point to the direction of maximum change?

@mahendrapanda4443 4 жыл бұрын

Please make a lec on real life application of matrix; projection of 3d image in eigen space and all that.

@ProfessorDaveExplains 4 жыл бұрын

check my linear algebra playlist

@Salamanca-joro 2 ай бұрын

4:10 if we are treating y^2 as a constant then why are we writing y^2? For example if we have this x^2(5) 5 is a constant so the derivative would be 2x since 5 is constant , and same goes for this question 4:10 , maybe it should have been 1 +3x^2 since y is constant? Instead of y^2+3x^2 I hope you understood my question

@AG-sq2dp 19 күн бұрын

Yeah, if it's supposed to be constant, I thought won't that become zero?! It did bug me for a while but then I understood that the key point is that even though y is treated as a constant when differentiating with respect to x, the y^2 term does not become 0 in the final PDE equation. This is because we are equating the two partial derivatives, not just looking at the derivative with respect to x alone.

@banderfargoyl 3 жыл бұрын

I have to admit that I've never understood why we have partial derivatives but not partial integrals. With the integral, the dx makes it clear which variable we're integrating and we don't need a special integral sign in addition.

@sudip7949 4 жыл бұрын

respect

@coolwinder 4 жыл бұрын

This is great, I have exam on 13th, can you make some more videos public :D

@GoBlue402 3 жыл бұрын

what do the i, j , and k represent in the grad f formula? (5:56)

@ProfessorDaveExplains 3 жыл бұрын

the unit vectors along the three axes

@GoBlue402 3 жыл бұрын

@@ProfessorDaveExplains So does the length of the vectors stay the same along each gradient. Sorry if I am not understanding something correctly

@carultch 2 жыл бұрын

@@GoBlue402 The vectors i, j, and k, are unit vectors that identify directions in the three cardinal axes of x, y, and z. Some books choose to hat the letters x, y, and z, to avoid a separate letter. It is an artifact of history that we call the unit vectors i-hat, j-hat, and k-hat. By definition, a unit vector has a magnitude of 1. This could be the axis unit vectors, or it could also be unit vectors in general. A unit vector's purpose is to identify direction, so it can give direction thru scalar multiplication to what otherwise would be a scalar quantity. Another application of unit vectors is in Newton's law of gravitation, and analogously, Coulomb's law, where r-hat is used as the radial unit vector, because the force acts radially along the line joining the two bodies.

@mathadventuress 4 жыл бұрын

I'm only in calc 2, and we barely started with differential equations... Interesting

@coxixx 3 жыл бұрын

is it true that gradient always points to summit of function?

@carultch 2 жыл бұрын

The gradient points in the path of steepest ascent. This could mean that it points to a peak of a function, and not necessarily the global peak of a function. It could simply point to a local maximum. It could also mean that it points to a saddle point of a function, where the function has two opposite curvatures meeting. It could also point to a local maximum on the function that is a continuous line, rather than a point-maximum.

@saritadalwani7847 2 жыл бұрын

Is advanced math platlist multivariable calculus ??

@vpa956 3 жыл бұрын

Explained it.

@aniketjoshi1610 4 жыл бұрын

Why are those videos private, in the playlist??????

@ProfessorDaveExplains 4 жыл бұрын

i release them one per week

@aniketjoshi1610 4 жыл бұрын

Hello sir! When are you going to release the video??????

@thevegg3275 4 жыл бұрын

Can someone help clear my confusion? When taking deriv wrt x of f(x,y), sometimes we say y is a constant so replace y with zero. Other times we say hold variable y as constant (and instead of replacing y with 0, we write down the y. This is so confusing!!! Here is clear example of my question. @t

@gunjanramteke909 4 жыл бұрын

I also noticed it

@gunjanramteke909 4 жыл бұрын

Please let me know if you find the solution

@alman5718 4 жыл бұрын

Suppose you have f(x) = x^2 + 2. When you find the derivative you will get f'(x) = 2x. The plus 2 is a constant and doesn't affect the gradient of the curve. Furthermore if you have a function for example f(x) = 4x^2. The constant at the front (in this case 4) will affect the gradient so doesn't cancel like adding a constant. So the derivative will be f'(x) = 4*2x = 8x. For partial derivatives of x all you're doing is treating y as a constant. just like the '+ 2' and the '4 * ' in the two examples. So let's suppose you have f(x,y) = xy + y. From the first example you take the ' + y' as a constant which it's derivative is zero since this won't affect the gradient. While the constant for 'xy' will stay. Making ∂f/∂x = y + 0 = y. Hope this helps.

@thevegg3275 4 жыл бұрын

Thanks so here is how I now see it. F(x,y)=x^2 + y. F'( x)= d/dx (x^2) +d/dx (y) =2x+0 F(x,y)=pi*x^2*y F'(x)=pi*y* d/dx (x^2) =pi*y*2x

@aiueo8962 6 ай бұрын

Why this is so easyy???? Thanks..

@idrissberchil25 4 жыл бұрын

6:00 you confused me there f(x,y) is a 3 dimension function taking 2 inputs, f(x,y,z) is a 4 dimensions function with 3 inputs. How does the grad vector get expressed in the previous 3d graph if we can't calculate the partial derv in z (df/dz) with the k unit vect.

@debarpan 4 жыл бұрын

Mr Booshit He probably meant a function that varies with three different variables (as in dimensions or axes).

@xOxAdnanxOx 4 жыл бұрын

Yes it’s like when you have a component in the z direction that you care about, they are all still 3D I think

@CROMast3r 4 жыл бұрын

f(x,y) is a surface in 3D, not the 3D itself

@AskAKill99 Ай бұрын

2 e to the to z (too easy)thanks for this very well explanation!!!!!

@trollthiti8045 5 ай бұрын

very good explanation i am from india/

@clkhaalaqtimir4677 Жыл бұрын

thanks professer dava i d,not more engish but iunderstand

@tanelkagan Жыл бұрын

Curious - what does using the "curly d" really add here? Could we not have done exactly the same thing using the standard d/dx, d/dy notation? What (I think) I am trying to say is that since we *know* we're dealing with a multivariable function, is it even possible that the standard d/dx (etc) notation could be misunderstood as referring to the "derivative of the whole function" even if that made any sense? The gradient sort of does that, so if we're looking at derivatives w.r.t. x and y, what do we gain in the intermediate steps by changing to "curly d"s? Or am I overthinking this, and we use curly ds purely as a label to remind ourselves that we're in a multivariable problem? Seems odd, should you need reminding if you're at this level of calculus!? 🤔

@kevconn441 4 жыл бұрын

Why do you say sometimes the derivative of the constant is 0, and the derivative of, say x, is x?

@ProfessorDaveExplains 4 жыл бұрын

the derivative of any constant is zero, and the derivative of x is 1

@kevconn441 4 жыл бұрын

@@ProfessorDaveExplains Thank you for the reply. I think my confusion is whether x is a constant in the original function or being held constant say if you are working out the partial derivative with respect to y.

@carultch 2 жыл бұрын

@@kevconn441 If you are taking the derivative of a function of multiple variables, relative to only one of the variables at a time, you treat all other variables as constants. So when using the d/dy operator, x becomes a constant in that particular differential operation. It is called a partial derivative when you do this, although the same principle still applies to differentiation in general.

@sauravprashar 3 жыл бұрын

I haven’t done derivatives in school yet so I am a bit confused that why is gradient of 2 variable function a 3d curve?

@sauravprashar 3 жыл бұрын

Ok got it we simply plot it in the 3d space like Z = f(x, y)

@ricardo.mazeto 4 жыл бұрын

Del? Aren't those called Nablas?

@ProfessorDaveExplains 4 жыл бұрын

i believe that is synonymous but outdated

@adityashankar5267 3 жыл бұрын

Finally, prof got a haircut 😂💇💇‍♂️

@kiliankraus 6 ай бұрын

I was geniuely so proud of myself that I could do the comprehension check lol

@kiliankraus 6 ай бұрын

thank-you for this video!

@yamatanoorochi3149 2 ай бұрын

product: u' v + v' u division: (u' v - v' u)/v² I find it easier to memorize like this u prime v plus v prime u has a ring to it

@simonediblasi8198 10 ай бұрын

That's a huge amount of knoweledge

@Siigrit Ай бұрын

These videos make me rethink my life choices. Uni is actually ass.

@prateekyadav9811 28 күн бұрын

what do you mean ?

@dinusiva3019 2 жыл бұрын

❤️❤️❤️❤️❤️❤️❤️❤️

@angelomartino4667 4 жыл бұрын

I love you

@mitschire 7 ай бұрын

i love you

@Imagon100 4 жыл бұрын

3:10 so if I have a curvy D does that make me a partial man?

@meharanas_43 4 жыл бұрын

So funny sir 😂😂😂😂

@MiltosPol-qn3zh 4 жыл бұрын

What are i, j and k ???

@ProfessorDaveExplains 4 жыл бұрын

ooh that's explained earlier in the series, check out the ones on vectors in my mathematics playlist, a bit before the calculus content starts, or possibly after calculus and before linear algebra

@DankFloyd-fe9bi 4 жыл бұрын

Unit vectors. It's a vector with a magnitude of one. These particular unit vectors point in the X, y and z directions and give you another way to notate other vectors. For example, you could write the vector as 2i+5j+4k

@MiltosPol-qn3zh 4 жыл бұрын

@Diogenes TheDog I understand almost 90% of calculus(you know what I mean, even some difficult multiple integrals and relatively difficult problems on them) and even the last advanced maths video proffesor dave uploaded but i find vectors really difficult to understand so i have many queries like this one