Partial Derivatives and the Gradient of a Function
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- เผยแพร่เมื่อ 2 ต.ค. 2024
- 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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"The gradient is kind of like a slope for a higher-dimensional function" is just what I needed to hear to conceptualize this. Thanks!
So good this explanation makes MIT lectures look like an overpriced DLC pack
🤣
ho can proffeser dave explains heart this comment?!
Man said "DLC Pack" lmao
same with Harvard lectures lmao
totally agree.. he explains algebra so nicely..i tend to sleep when i listen Glibert's lecture :-)
Gradient: A vector made up of all the partial derivatives of the function. Thank you!
Wtf is Jacobian then??
Pendulum theSimpleOne lmk when you get an answer
@@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.
@@Darkev77 he got the answerrrrr seeee
@@ankitaaarya beast
This is awesome. Feel like I finally understand the gradient now.
Best 10 minutes and 56 seconds of my life. Such a clear explanation!
Fuck Intercourse, watching Dave is the real deal ❤
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.
That's cool to know. Thanks.
Thanks to you too Professor Dave.
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.
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)
This man carried me through general chemistry, calculus II and now calculus III. I can't thank you enough for teaching me math.
An eye opener video. Neat and tidy.
You rock man! gracias amigo. Im so happy to finally understand it
Was the video time 11 mins😲I didn't have a feeling that 11 mins have passed by. it was deeply interesting.thanks sir🤗
happy teacher's day sir.....
Do you teach all subjects?👍good work btw....
where is "he knows a lot about science stuff" :(
Okay the content of this video is super but the Intro always have me like 😂😎
Just got to see ur videos sr..... U are an super osm educator... Lots of love ❤😘
2 e to the to z (too easy)thanks for this very well explanation!!!!!
Thankyouuu soo much professor….absolutely incredible explanation!!!!!
Thank you sir!
I wish you to make vedio on Total differentiation.
Please !
Please!
Please!
Please!!!!!!!!!
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.
I watch this channel so much that I once had a dream and your theme song made a cameo 😂
Just awesome !!!
But I don't understand why x axis + y axis vector will point to the direction of maximum change?
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.
Great explanation! Just one question: Why does the gradient point in the direction of maximum slope?
due to the addition of the partial derivative vectors using laws of vector addition
@@fineartpottamus9020 this was the final puzzle piece for me, now it all clicked together. thank you a lot.
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.
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
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.
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)
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.
I'm only in calc 2, and we barely started with differential equations... Interesting
Hallelujah, just saved my calculus...
I just went down and liked every comment that was positive on this video. Its the only way to extend my appreciation! Thanks man!
Great teacher
Oh thank you so much! After 19 years I can finally picture it!!!
Finally, prof got a haircut 😂💇💇♂️
Wow great explanation. Sucks to see us students pay for an education where profs have a hard time explaining clearly. Thank You!!
Please make a lec on real life application of matrix; projection of 3d image in eigen space and all that.
check my linear algebra playlist
very good explanation i am from india/
God!!!!!
You saved me !!!!!
I have test tomorrow on this topic !!!!
Thank you so much. Finally understood the real meaning of partial derivative and gradient. 😇
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 :)
thank you so much you are the best
Del? Aren't those called Nablas?
i believe that is synonymous but outdated
Thank you....
happy teacher's day
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 TH-cam Channel.
Sir i Thank you & Salute You.
What a Life saver, thanks so much, professor
this is what you call a video. One of the best teaching videos I've ever watched
thank you sir.!!!!!!
I love how clear and concise your explanations are!
This is highly underrated stufffff
Ngl I’m lucky i found this 🔥 I subscribed❤️
Awesome
This is very helpful.
Why don't you do a video on higher order partial derivatives and total differention
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
I also noticed it
Please let me know if you find the solution
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.
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
Wow You have explained very good Finally I understand this concept Keep it up.
I couldnt get z
Great explanation, thanks! However, I didn't quite understand what i, j, and k represent in the equations around the 6th min
Why do you say sometimes the derivative of the constant is 0, and the derivative of, say x, is x?
the derivative of any constant is zero, and the derivative of x is 1
@@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.
@@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.
Del means gradient. Thank you for clearing that up for me.
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
This is great, I have exam on 13th, can you make some more videos public :D
lifesaver
Thank you sir for your dedication and for making this free! 🙏
Really helpful video, thanks so much :)
wow, amazing video. please keep making more ML videos!
These videos make me rethink my life choices. Uni is actually ass.
what do you mean ?
🤣🤣🤣
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...😇
that's the next one!
Thanks for making that so simple for me @ 59. Cheers!
thanks professer dava i d,not more engish but iunderstand
Thanks sir, clarified. Please do on line integrals and Greens theorem..
those are coming!
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?
Ok got it we simply plot it in the 3d space like
Z = f(x, y)
I love you
That's a huge amount of knoweledge
Very clear explanation thanks
respect
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!? 🤔
Good explanation sir
Why this is so easyy???? Thanks..
Excellent explanation!
Thank you very much! It is the best explanation of partial derivatives that I ever heard!
I was geniuely so proud of myself that I could do the comprehension check lol
thank-you for this video!
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.
is it true that gradient always points to summit of function?
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.
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.
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.
Mr Booshit He probably meant a function that varies with three different variables (as in dimensions or axes).
Yes it’s like when you have a component in the z direction that you care about, they are all still 3D I think
f(x,y) is a surface in 3D, not the 3D itself
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.
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.
Why is all of this less difficult for me than understanding implicit differentiation
i,j,k ???? Why maximum (!?) rate of change? Thats the essence. You should‘nt teach differential rules. Thats another thing and has nothing to do with the gradient, its just technique. So, no.
The letter trio of i/j/k indicates the three coordinate unit vectors, that correspond to x, y, and z directions respectively. If you're curious to know why we use that particular trio of letters, look up Hamilton and quaternions. He had wanted his concept of quaternions to replace how we use vectors today (and in some applications, it has). It didn't catch on, but much of the notation still did. That's a topic for another day, but that is the origin of why we use that particular trio of letters. I prefer x hat, y hat and z hat instead.
I don't understand the rest of your comment.
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?
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!!!!!!!!
Is advanced math platlist multivariable calculus ??
Nice review of gradients! Thanks!
Thank you.
What are i, j and k ???
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
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
@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
Sequence of watching videos is troubling me , content is way too good but not able to see in Sequence 😭😭
Just go to the mathematics playlist.
i love you
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
2 millions
the coolest and prettiest explaining teacher I've ever seen!
Literally the best explanation. Trying to do a project where partial derivatives come up, and I needed a quick refresher.
3:10 so if I have a curvy D does that make me a partial man?
Why are those videos private, in the playlist??????
i release them one per week
Hello sir!
When are you going to release the video??????
❤️❤️❤️❤️❤️❤️❤️❤️
Clear as water, helps me understanding Deep Learning!