Partial Derivatives and the Gradient of a Function

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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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3 окт 2024

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Комментарии : 170

@adityaadit2004 4 года назад

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

@SumriseHD 6 месяцев назад

Fuck Intercourse, watching Dave is the real deal ❤

@KidDroskii 4 года назад

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

@davidbowman9695 4 года назад

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

@maxcl3474 3 года назад

🤣

@aashsyed1277 3 года назад

ho can proffeser dave explains heart this comment?!

@samsonlawal1600 3 года назад

Man said "DLC Pack" lmao

@saf8514 2 года назад

same with Harvard lectures lmao

@hemantsharma17 2 года назад

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

@onesun3023 4 года назад

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

@ycombinator765 4 года назад

Wtf is Jacobian then??

@Darkev77 4 года назад

Pendulum theSimpleOne lmk when you get an answer

@tejaswinikarpe3911 3 года назад

@@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 года назад

@@Darkev77 he got the answerrrrr seeee

@Darkev77 3 года назад

@@ankitaaarya beast

@strugglingcollegestudent 2 года назад

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

@radiacia_3511 11 месяцев назад

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)

@kahdargo7 5 лет назад

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

@sreedhar75perupally 2 года назад

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 RU-vid Channel. Sir i Thank you & Salute You.

@StephenGillie 5 лет назад

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 11 месяцев назад

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

@GAment_11 4 года назад

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

@idealthinker101 2 года назад

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.

@danielcheong4804 2 года назад

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

@pulpettedilmare9597 4 года назад

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

@jackanderson8278 3 года назад

I love how clear and concise your explanations are!

@josephabboud1151 2 года назад

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 :)

@goutsugoutsu 4 года назад

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

@alejandroalmarza8447 2 года назад

Profesor You are amazingly clear…like all my colleagues here say your <a href="#" class="seekto" data-time="657">10:57</a> seconds video summarized hours of calculus classes.. my admiration to you. Thanks

@Kiky_MedPhysicist Месяц назад

Thank you sir for your dedication and for making this free! 🙏

@buraxta_ 2 года назад

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

@te-kowski Год назад

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

@Deepak-pi9xx 3 года назад

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

@qualitytoolbox4872 4 года назад

An eye opener video. Neat and tidy.

@gandalfthegaytwotowerdestr3391 4 года назад

What a Life saver, thanks so much, professor

@elenaroyss7810 8 месяцев назад

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

@Borntowin894 4 года назад

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 5 лет назад

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

@devanshujoshi8393 3 года назад

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

@sureshtanwar3588 5 лет назад

happy teacher's day sir.....

@antonbreugel3332 4 года назад

Hallelujah, just saved my calculus...

@JoshVandever 20 дней назад

Del means gradient. Thank you for clearing that up for me.

@amanjmullick2930 5 лет назад

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

@ryannkohlman5751 3 года назад

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

@theologyscienceandpropheti6808 5 лет назад

Thank you.... happy teacher's day

@kaneezfatima926 4 года назад

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

@Dennis4Videos 3 года назад

Clear as water, helps me understanding Deep Learning!

@pusheletsommatladi4686 4 года назад

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

@sotiris41664 2 года назад

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.

@abhradeepghosh7102 2 года назад

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.

@elharithhashim4424 5 месяцев назад

Very clear explanation thanks

@MohammadBenSalamah 4 года назад

Excellent explanation!

@sstein5866 4 года назад

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

@fineartpottamus9020 4 года назад

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

@carmelwolf129 2 года назад

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

@carultch Год назад

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.

@aniketjoshi1610 5 лет назад

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

@coolwinder 5 лет назад

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.

@elizabethsimakando7299 3 года назад

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

@MinhLe-xk5rm 4 года назад

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

@HeathWatts 11 месяцев назад

Nice review of gradients! Thanks!

@andrii.kukuruza 15 дней назад

Great explanation, thanks! However, I didn't quite understand what i, j, and k represent in the equations around the 6th min

@nancysanskriti2158 3 года назад

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

@AskAKill99 3 месяца назад

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

@FD-rt3rv 2 года назад

Fantastic explanation

@88NA Год назад

Thank you Professor Dave

@shlokekhullar4261 2 года назад

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

@vikramnagarjuna3549 5 лет назад

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

@ProfessorDaveExplains 5 лет назад

those are coming!

@sambananas4513 5 лет назад

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

@con_el_maestro3544 Год назад

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

@rigbyb 4 месяца назад

Really helpful video, thanks so much :)

@portgasdace8961 5 лет назад

Just awesome !!!

@abdullahalaraz7404 4 месяца назад

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

@simantajenaadvancedmathema9764 5 лет назад

Good explanation sir

@banderfargoyl 3 года назад

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.

@mrhatman675 4 года назад

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!!!!!!!!

@Salamanca-joro 5 месяцев назад

<a href="#" class="seekto" data-time="250">4:10</a> 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 <a href="#" class="seekto" data-time="250">4:10</a> , 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 3 месяца назад

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.

@powercables 2 дня назад

when taking the derivative in parts you treat y as a constant because your deriving with respects only towards x. say you derive 3x then answer is 3 same with y * x dx (dx = with respects to x) = y. So when taking the derivative of y^2 * x dx you get y^2. if it was y^2 * x^2 dx you would use the power rule only for x becuase your treating y as if its numerical so you would get 2 * y^2 * x. I hope this helped! (sorry if it was confusing)

@powercables 2 дня назад

also you are right, when deriving a constant without x attachted its 0 so in the equation x^2 + 2xy - y^2 deriving with respects to x you would get 2x + 2y - 0 becuase the y^2 is like a number and the derivative of a number without a pronumeral (that you are treating as a pronumeral not a number so not y in this case) is 0.

@asaidinesh5220 5 лет назад

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 5 лет назад

that's the next one!

@mathadventuress 4 года назад

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

@namelessbecky 9 месяцев назад

Thank you.

@ahmedelsabagh6990 4 года назад

Great teacher

@ZYau-lc5ql Год назад

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 Год назад

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.

@giorgosrallis7044 3 года назад

Great video

@mahendrapanda4443 5 лет назад

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

@ProfessorDaveExplains 5 лет назад

check my linear algebra playlist

@trollthiti8045 7 месяцев назад

very good explanation i am from india/

@yamatanoorochi3149 5 месяцев назад

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

@naders. 2 года назад

Thank you! 😊

@adityashankar5267 3 года назад

Finally, prof got a haircut 😂💇💇‍♂️

@kaan7120 3 года назад

thank you so much you are the best

@ricardo.mazeto 5 лет назад

Del? Aren't those called Nablas?

@ProfessorDaveExplains 5 лет назад

i believe that is synonymous but outdated

@coolwinder 5 лет назад

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

@aiueo8962 8 месяцев назад

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

@omer7895 2 года назад

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?

@ayushagarwalroll0283 3 года назад

thank you sir.!!!!!!

@samurainair1 4 месяца назад

Awesome

@vpa956 4 года назад

Explained it.

@kiliankraus 8 месяцев назад

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

@kiliankraus 8 месяцев назад

thank-you for this video!

@sudip7949 4 года назад

respect

@santinacasari311 Год назад

Valeu!

@idrissberchil25 4 года назад

<a href="#" class="seekto" data-time="360">6:00</a> 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 года назад

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

@xOxAdnanxOx 4 года назад

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 года назад

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

@thevegg3275 4 года назад

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 года назад

I also noticed it

@gunjanramteke909 4 года назад

Please let me know if you find the solution

@alman5718 4 года назад

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 года назад

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

@kevconn441 5 лет назад

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

@ProfessorDaveExplains 5 лет назад

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

@kevconn441 5 лет назад

@@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 3 года назад

@@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.

@tanelkagan 2 года назад

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!? 🤔

@Siigrit 4 месяца назад

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

@prateekyadav9811 3 месяца назад

what do you mean ?

@Kiky_MedPhysicist Месяц назад

🤣🤣🤣

@scrambledsocks9295 Год назад

where is "he knows a lot about science stuff" :(

@jaychunny2102 День назад

lifesaver

@clkhaalaqtimir4677 2 года назад

thanks professer dava i d,not more engish but iunderstand

@simonediblasi8198 Год назад

That's a huge amount of knoweledge

@ricardosefa4186 Год назад

I couldnt get z

@jamespatrick9191 3 года назад

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

@kashyaptandel5212 Год назад

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

@sauravprashar 3 года назад

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 года назад

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

@saritadalwani7847 3 года назад

Is advanced math platlist multivariable calculus ??

@MiltosPol-qn3zh 5 лет назад

What are i, j and k ???

@ProfessorDaveExplains 5 лет назад

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 5 лет назад

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 5 лет назад

@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

@coxixx 4 года назад

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

@carultch 3 года назад

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.