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Me in highschool math class: never pays attention Me in bed at 3am: Yes Derivative is and has always been my passion

@fufaev-alexander

9 ай бұрын

haha :D

and i thought it was just a cooler way to write...

Well, in this case the difference is easy: The partial derivative is expected to use sensible notation, whereas the total derivative somehow thinks that y is a function.

@simonmarcoux5879

Жыл бұрын

if y was independent of x and you still used d instead of del ...it would yield the exact same result. I wholly agree with your opinion that it is by no mean a sensible notation nor is it intuitive.

@artey6671

Жыл бұрын

@@simonmarcoux5879 I once took a class where I worked on the problem sheets with a physicist. He understood the strange notation and he tried to explain it to me, but I didn't get it. I gave up on the class not much later and he apparently almost made it at least. It was semiclassical analysis.

@BladeOfLight16

Жыл бұрын

It's not a question of what the operation "thinks." It's a question of the _assumptions_ *you* are imposing on the problem space. A partial derivative means you are assuming the variables vary independently, and this means the derivative of the variable with respect to the other variable is 0. y does not change value _at all_ when x changes value; any pair of values for x and y is possible (within their respective domains). y can still be a function, but it cannot be a function _of x._ A total derivative discards this assumption and allows for the possibility that they may be dependent, but it reduces to the partial derivative in the case when they are not. It is not valid to take a partial derivative when the assumption of independence does not hold. In other words, taking a partial derivative is declaring that you know (or are assuming) the variables to be independent. It would be completely equivalent to merely write, "Because x and y are independent, dy/dx = 0," and use that to continue solving the problem instead.

@artey6671

Жыл бұрын

@@BladeOfLight16 Is that what engineering calculus books say? Damn.

@simonmarcoux5879

Жыл бұрын

@@BladeOfLight16 wow! Now that is what i call a clear explaination. It lift away the confusions I had while watching the video. I must admit that it as been 12 13 years since the last time i used any form of partial/total derivative in a math sence. I still often calculate a discrete rate using embedded programming, but it has 0 mathematical formality to it: change on a value divided by the elapsed time between values. Thanks a lot for the insightful reply.

I love that this video is so short and yet insightful.

I’m honestly surprised they never explained this when I took multivariable calculus.

@moslynmoslyn679

2 ай бұрын

seriously😅

Basic stuff but excellent explanation!

@fufaev-alexander

Жыл бұрын

Thank you! Do you like short videos like this that answer a specific question?

@mathematicaleconomist4943

Жыл бұрын

@@fufaev-alexander Brilliant! I am generally more likely to "tap on" brief videos (like this one!). There's usually no time! The more intuitive the explanation, the better! That being said, I am certainly not allergic to long videos when I have the time. But I definitely like the brief ones!

@kikilolo6771

Жыл бұрын

@@fufaev-alexander Personally I like short videos like these that make things clear and don't go into too much stuff we don't care but I also appreciate longer videos that go more deeply into the subject

@Flaystray

Жыл бұрын

Yes, because everyone in the world should already know this by now! If they don't, they must be fucking stupid!

@TepsiMorphic

Жыл бұрын

​@@fufaev-alexander i also prefer short videos but the thing is, can you always compress an interesting topic into a short video without it losing something meaningful?

Basically this: if partial, treat all other variables other than the one you’re differentiating with respect to as a constant. If total, differentiate everything.

I needed this video in my Differential Equations course back in 2014. Thanks dude!

I appreciate the brevity of the video. It takes a while for concepts, even basic ones, to sink in.

super clear and quick explanation + nice little exalple to understand everything. Great video!

this video gives better understanding. nice explanation! thank you so much for clearing all confusion.

I only learned about this today and it's not even in my curriculum yet Excited about that

Okay in simpler terms Partial derivative differentiates the x term normally but expects every other term to be a constant.

As a student I loved loved differential and partial differential equations. They were the poetry of engineering. They represented beauty and truth.

@con_el_maestro3544

9 ай бұрын

I need to know what you drinking, I guess my stuff is not strong enough

@phild8095

9 ай бұрын

@@con_el_maestro3544 I'm drinking Iowa water with a touch of lemon juice. Worcester Polytechnic Institute. Dr. John Van Alstyne taught ordinary differential equations and Dr. Robert Wagner, Dr. Anthony Dixon, Dr Y.H. "Ed" Ma, Dr. Bob Thompson and Dr Al Sacco all taught Chem Eng Partials in various classes. The best was a project I did with Ed Ma. It was modeling physical adsorbtion in 3d and finding the speed of molecules diffusing into pores of zeolites. I developed about 20 equations and worked it down to the one that gave velocity of molecules as they bounced down the tine pores in the zeolites and then plugged in my raw data to get the diffusion rate and mean velocity. Absolutely glorious. That was many decades ago. About 15 years ago in the process of moving I found my research project report and started going through the derivation and was amazed at what I had done my senior year. That report was the most hard core engineering math I had ever done. I suppose it is still in the WPI library as a pdf. As part of my research to do the project I had done some reading and found a mistake in some work that had published a few years prior. I don't miss the endless hours solo in the lab, but the writing and modeling was a blast.

Short and precise. Thank you for this!

As I understand it the main motivation behind the difference in notation is that it serves as a preventative against certain careless errors, for example it reminds you that you can't use the two-dimensional version of the chain rule.

Wish i found the channel when you started 3 years ago, it would have been a huge help to me

Wow 4 years of engineering and 3 thermo and thermo related modules, and I find out it was that simple

Great stuff, Sir keep it up.

Thanks this helped a lot. I am currently studying Potential energy and Force so this helps a lot in making out the difference between both the types of derivatives

Exactly the thing i needed to understand my mechanic lecture , thank you a lot!!

@fufaev-alexander

Жыл бұрын

Well, I'm glad I could help you make sense of your mechanic lecture! Keep up the mechanical brilliance!

Best video explaining the difference. Good job.

Straightforward explanation, thank you sir

@fufaev-alexander

Жыл бұрын

Hey Yaren, thank you for your feedback!

1:43 very well spent. Thanks very much!

awesome video! thank you!

Interesting, I had only ever seen them as separate notations for the same operation.

thank for this video. It's very easy to understand compare to my lecture

There’s a deficiency in notation for partial derivatives. They should strictly be followed by a vertical bar and a subscript to indicate what is being kept constant. Normally this is taken for granted, but it can cause confusion.

@death_parade

Жыл бұрын

Thanks for mentioning this.

@zeybarur

Жыл бұрын

Yes, it's extremely important in thermodynamics, for exmaple... Learned that the hard way 😅

@egoreremeev9969

Жыл бұрын

Actually there is no deficiency in notation in the case of partial derivatives. Each other variable is by definition of partial derivative - constant, so notation \partial_{x_2} f(x_1,x_2,...,x_n) is telling us how function behaves on line (x_1, y, ..., x_n) with y - as variable It only becomes necessary if we want to make "not total" derivative, i.e. keeping some variables constant.

@iyziejane

Жыл бұрын

The reason we do that in thermodynamics is not that partial derivatives are unclear, but to remind readers of which variables are taken to be the independent variables for each of the thermodynamic potentials. So the thermodynamic energy E is naturally a function of entropy S and volume V (so that pressure P and temperature T are not independent variables, but depend on S and V). The Helmholtz free energy F = E - TS naturally trades out entropy S for temperature T. E(S,V) vs F(T,V). The Legendre transform is used to switch one independent variable for another because of the way it leads to the the corresponding differentials being changed, e.g. dE = T dS - P dV compared with dF = - S dT - P dV . The extra notation on the partial derivatives is just an extra helper to keep these definitions straight, so you may write T = (\partial E / \partial S)_V to remind you that V is the other independent variable when your thermodynamic potential is E. But that fact never changes and can just be learned as part of the definition of E.

@sebastian8870

Жыл бұрын

That is common notation in thermodynamics!

Man I’ve been looking for this video for the past 6 years.

...gerade habe ich eine KZread-Synopsis zum Thema gewöhnliche Differentialgleichungen gesehen und weil Differentialgleichungen für mich ähnlich schön sind wie klassische Musik, kommt hier meine Kurzsynopsis zum Thema, wie ich sie unterscheide: eine gewöhnliche DGL ist jede DGL, die nicht partiell ist, was bedeutet, dass in ihr nur nach eine Größe abgeleitet wird... z. B. ist die Funktion z = f ( x, y ) nach mehreren Größen ableitbar, sodass sie partiell ist, und eine Ableitung ist z. B. df ( x, y ) / dx, wobei man sich das bitte in kyrillischer Schreibweise vorstelle, weil es die partiellen Ableitungen indiziert. Eine DGL ist linear, wenn die Größe, nach der abgeleitet wird nicht quadratisch ist, keine Wurzel vorhanden ist und kein Winkelargument vorkommt, zudem ist y ( x ) mal y' ( x ) = 1/ x nichtlinear, weil die Ableitung mit der gesuchten Funktion multipliziert wird, was Linearität auch unmöglich macht. Weiterhin gibt es lineare Differentialgleichung mit konstantem Koeffizienten, was die Koeffizienten betrifft, die vor x stehen, indem sie konstant sind, wenn sie nicht von x abhängig sind. Homogen ist eine DGL, wenn die Größe, nach der abgeleitet wird oder eine ihrer Ableitungen in allen Termen vorkommt, sodass jeder sieht, dass y''( x ) - 3y' ( x ) = 2xhoch2 inhomogen ist, weil der Term rechts der Äquivalenz eine Funktion ist, die für die Inhomogenität sorgt. Die höchste Ableitung schließlich bestimmt die Ordnung einer DGL, wobei das für viele ein Stolperstein bei zweiten Ableitungen ist, indem sie fälschlich annehmen, das Quadrat zeige Nichtlinearizität an, wobei ich diesen Flüchtigkeitsfehler vermeide, indem ich mir die Terme geklammert denke, sodass sich die Linearität klar zeigt, weil das Quadrat eben sehr deutlich einer Ableitung zugeordnet werden kann, die Teil eines Term ist. Für mich fast so schön wie Musik von Bach... ...übrigens ist das Erkennen von Unterschieden in Differentialgleichungen nicht wirklich schwierig, aber sehr inspirierend ( ...es erfüllt mich mit Leben... ), und es ist unerlässlich, weil es für die verschiedenen Arten von DGL's auch verschiedene Wege gibt, sie zu lösen... Le p'tit Daniel

Partial derivative simply means : all other variables are kept constant. For two variables it is also useful to know the geometric interpretation: z= f[x,y] represents a surface over the (x,y)- plane . If we take the partial derivative with respect to x , we get the slope of the curve obtained by intersecting the surface and the plane y = constant , and of course similarly if we take the derivative with respect to y .

@solwizard

10 ай бұрын

Thanks! You nailed it for me.

@vladimirsemenyuk792

9 ай бұрын

Finally, someone who knows math

nice video!

How did you create the animations for the video? I love them!

For someone who wants to know more, total derivative is only used when there is only one input (in Real), i.e. a path composed with the function. To be more specific, let f(x1,…xn) be a real valued function, and g(x) =(g1(x),g2(x),…,gn(x)) be a vector valued function, then the function f o g is a single variable function that has one input, we call the function g a path. So when the path is specified, we write d/dx f = d/dx (f o g), where the (total) derivative is clearly well defined for f o g. The path makes all other variable dependent on one. Intrinsically there is no relation between each variable x1,…,xn, it is the entries of the path g that has relations. Note that for one variable functions, partial derivative is the same as total derivative, because they are just normal derivative (not in the sense of normal direction derivative).

Thank you very much sir! very useful video

@fufaev-alexander

Жыл бұрын

Thank you for the comment! 🙃

thank you. I learned this many years ago but couldn't remember the difference. Now I know again.

That's really helpful!

Partial differentiation are the mini major bosses you see in games that very scary but easy to beat

This is the clearest explanation I've seen

@fufaev-alexander

Жыл бұрын

thanks, tolkienfan1972!

Interesting to note that, if in doubt, you can use the second method: the first one is just one step further...

Wow, I took this on Calculus so long ago that I just realized that I forgot something so simple. I got both wrong in the _y_ part. Keep practicing your math or it will go away.

Thank you amazing video

Omg never knew maths was this interesting!

Many students would simplify both to f/x😲

All I know is that when the ratio of prices in a budget is equal to the ratio of partial derivatives of the Indifference Curve of the items in that budget, that’s the amount that maximizes utility.

Short and to the point.

@fufaev-alexander

Жыл бұрын

Thanks! Check out my other videos as well! :) Or my website: en.universaldenker.org

thank you for the information

@fufaev-alexander

Жыл бұрын

thank your for the comment

thank you

Thank you!

@fufaev-alexander

Жыл бұрын

Thank you back, for the comment, Evelyn!

practically partial deriviation is just a special case of total deriviation where y(x) is constant

@fufaev-alexander

10 ай бұрын

yep

In the total differential, not that y can depend on x, but because the dx, and dy are considered as an elementary objects, so in d(3x^2)/dx we can write d(3x^2) = 6xdx than the dx cancel out, but in the case of dy/dx, dy can not be cancelled with dx, both are considered as an elementary differential objects in a defrent axis

I took college math all the way through partial diff eq and I still didn’t know the difference until 10 seconds ago

I usually dislike using partial and total notation for the same function. It becomes very dependent on context and interpretation. Having g(x)=f(x,phi(x)) leads to unambiguous dg/dx notation that can also be expanded wrt the partials of f, but notation overkill can be a danger too. One of the things that is hard to explain in a chain rule is when the variable name appears at two different depths, when a truly unambiguous chain rule would have different notations for each variable slot of each function, but again, the price is notation overkill. Overall it kind of sucks, lol.

Gracias chavall!!!

Opps I have been using them both interchangeably not knowing they are different💀💀💀💀💀💀💀💀

Beautiful explanation.

love it❣

@fufaev-alexander

Жыл бұрын

Thank you for your positive feedback, Jafir!

thanks bro

Maybe I'm missing something here but this doesn't make any sense? For a single variable function of course the partial and total derivative are the same. When we have a multivariate function (I'll stick to a vector space like R^n cause I'm not that well informed for other situations), we can take partial derivatives in each variable and also in some linear combination of them (a directional derivative). The relevance being of course that this derivative is the standard derivative of the function at the point we evaluate at, in "the direction of" the variable we differentiated by. The total derivative (if it exists, which it might not) is then a function which encodes all this information for all possible directions at all points, which, in the case of a multivariate function, will lead to this function being matrix valued (in the case f:R^n->R^m, it will be a mxn matrix), with each entry of the matrix a function of the original variables. Overall, the total derivatives represents the tangent plane, whereas the partial derivative just tells you the slope of that plane in a particular direction. Of course, there is a relation between the two derivatives: we can obtain partial derivatives by applying this total derivative linear map to the direction we want to consider and we can obtain the total derivative by considering the partial derivatives in the direction of each of the variables to get a column vector for each. We then combine these columns into a matrix and we get our total derivative. It's important that any direction vectors we use are normalised so we don't create random constants everywhere. Sometimes we write this total derivative as Df, or maybe ∇f, but writing df/dx for a function f(x,y) just seems wrong, unless y is some function of x itself, in which case the partial derivative should also acknowledge this. The partial and total derivatives for a function of the form f(x,y(x)) will be the same, since this f is really just another function g(x)=f(x,y(x)). When we compute the total derivative, it will be a mx1 matrix (since g is R->R^m), and this corresponds to the column vector we get when we take the partial derivative in x. I come from a pure maths background and some people in the comments seem to be physics based, so perhaps this is some sort of weird notation physics is using, but this is not the difference between total and partial derivatives.

@t33can

Жыл бұрын

This is the actual answer. Like you said, the video is somewhat confusing and doesn't seem to realize that both expressions as stated in it must lead to the same result.

@iyziejane

Жыл бұрын

The videos notion of total derivative is related to your more sophisticated perspective of the total derivative as a linear map. First we are specializing to functions f: R^n -> R so your total derivative that is matrix valued in general is now a single row vector, (i.e. the co-vector that represents the linear functional, or it is also a differential form). I'll write Df = [ (∂f/∂x_1) , ... , (∂f/∂x_n)] to represent this linear map as a row vector. Now suppose you have a smooth function X: R -> R^n with X(t) = (x_1(t),...,x_n(t)). Let F be the composition of f with X, so F(t) = f(X(t)). Now dF/dt is a derivative of a real-valued single variable function. We can express it with the chain rule as (d F/dt) = (∂f/∂x_1)(dx_1/dt) + ... + (∂f/∂x_n)(dx_n/dt). In these introductory calculus classes, this chain rule expression for (d F / dt) is called a total derivative. As you can see, it is your total derivative linear map acting on the unnormalized tangent vector to the curve X: R -> R^n at that point. This situation comes up all the time in calculations, including physics, or anything to do with real time. To take a finance example, the price of A might be related to prices of other goods B, C , D, so we have P_A(B,C,D) , but all these prices are functions of time as well, so d P_A / dt will use this "total derivative" chain rule.

@JoseGalois

9 ай бұрын

you are right. they all have no idea what's happening and are just playing around with notation.

Found helpful 🎉

So the partial derivative is when dy/dx=0?

@matusmoro955

10 ай бұрын

Yea

@austinburrington6434

2 ай бұрын

In this example it’s a consequence of the partial derivative. I like to think about it in terms of orthogonal systems.

Thank you sir

@fufaev-alexander

11 ай бұрын

Thank you too! And welcome to the World of Physics :)

wow! nice video

@fufaev-alexander

Жыл бұрын

thanks!

Very nice

If you could explain with some applications, it could be useful to understand need of these derivatives.

Hey!! Great video! I am very new to calculus. Can you explain how y depends on x in complete derivative?

@yarenkaya7872

Жыл бұрын

It needs to be already given or inferred. To exemplify, if you're in a circle with radius r centered at origin, you know y = +/- sqrt(r^2-x^2). Then, you know dy/dx is 2x/sqrt(r-x^2).

@BIOLOGYWORM

Жыл бұрын

❤ty

@yarenkaya7872

Жыл бұрын

@@BIOLOGYWORM You're welcome

I never thought I would understand partial derivatives. But now maybe I do.

Thanks for the explanation. I was so confused, I had to go on a site to get help, and all the people there did was try to tell me I know nothing. Yeah no shit that's why I'm asking. Anyway thanks for the video it really helped.

Fantastic

If only my calculus teacher had explained it this simply 😑

In partial derivative, y is seen as constant and derivative of any constant is zero.

Beautifully explained. Thank you! 😊

@fufaev-alexander

Жыл бұрын

Thank you for your positive feedback!

But if you do a total derivative of x in a circle, how in y dependent on x?

so, the only difference (by formal) is whether you keep the dy/dx (where top & bottom are different variables) terms?

I like to think of a partial derivative as taking the total derivative of curve thats formed when u cut an n dimensional function with a 2d plane. The partial derivative would be the slope of the tangent line in that confined 2d space.

u r king

@fufaev-alexander

Жыл бұрын

Thank you, Ismail 🤓

Does this makes a difference in calculation?

So nice thanks

@fufaev-alexander

Жыл бұрын

Thank you for the comment!

The difference is the difference between partial y contained in x and the other way round where the answer must be derived again.

My man being a better teacher in 2 minutes than my analysis teacher in 2 months

Is there a limits equivalent for partial derivatives?

Simple its not the whole infentesimal change in f just the part in one direction. Thats why when you learn directional derivatives its just adding them together to get the actual change.

What's the value of d(dx)/dx ? Is the same of ∂(dx)/∂dx ?

Here in India we teach This as Follows Next, an excerpts from one of my books: This fourth example is regarding the differentiation of functions in two variables. For a function in two variables, say z = z(x, y), at a value x=a, y=b, the elementary change in z, say dz, can be seen in infinitely many directions. In fact, these directions are along the lines at various angles starting from the point (a, b) in x-y plane. Let us consider an arbitrary direction from the point (a, b) at an angle, θ. Angle θ is measured anticlockwise from the increasing direction of x. We trace an elementary distance, say dr, on this line from point (a, b). At the end of this elementary trace, the coordinates of the point specify the respective changes in values of x and y. The term “dr” refers to the elementary change in a variable r that takes values as the distances along the direction θ. Obviously, dx will be dr.cosθ and dy will be dr.sinθ. Hence, tanθ = dy/dx and dr is √[1 + (dy/dx)2].dx. Now, at point x= a, y= b, in this direction, that is at angle θ, the change in z, that is dz, is z(a + dx, b + dy) - z(a, b). Let us see the rate of change in this arbitrary direction, θ. Rate of change here can be considered in three needful cases- 1. When the variable with respect to which the rate is calculated is r, the rate of change is called directional derivative. It is [z(a + dx, b + dy) - z(a, b)]/dr. It is denoted as dz/dr. 2. When the variable with respect to which the rate is calculated is x, the rate of change is called x-derivative. It is [z(a + dx, b + dy) - z(a, b)]/dx. It is denoted as dz/dx. 3. When the variable with respect to which the rate is calculated is y, the rate of change is called y-derivative. It is [z(a + dx, b + dy) - z(a, b)]/dy. It is denoted as dz/dy. Now, at the point (a, b), we define the directional derivatives in two standard directions- 1. Along x-axis, θ=0, we define [z(a + dx, b) - z(a, b)]/dx, which is called partial x-derivative. It is denoted as ∂z/∂x. 2. Along y-axis, θ=Л/2, we define [z(a, b + dy) - z(a, b)]/dy, which is called partial y-derivative. It is denoted as ∂z/∂y. Now, we can express dz/dr, dz/dx and dz/dy in terms of partial derivatives- 1. A re-arrangement of [z(a + dx, b + dy) - z(a, b)]/dr gives- [z(a + dx, b + dy) - z(a, b + dy)]/dr + [z(a, b + dy) - z(a, b)]/dr = [z(a + dx, b + dy) - z(a, b + dy)]/[ √[1 + (dy/dx)2].dx] + [z(a, b + dy) - z(a, b)]. [dy/dx]/[ √[1 + (dy/dx)2].dy] = [∂z/∂x]/ √[1 + (dy/dx)2] + [∂z/∂y].[dy/dx]/ √[1 + (dy/dx)2] 2. A re-arrangement of [z(a + dx, b + dy) - z(a, b)]/dx gives- [z(a + dx, b + dy) - z(a, b + dy)]/dx + [z(a, b + dy) - z(a, b)]/dx = [z(a + dx, b + dy) - z(a, b + dy)]/dx + [z(a, b + dy) - z(a, b)]. [dy/dx]/dy = [∂z/∂x] + [∂z/∂y].[dy/dx] 3. A re-arrangement of [z(a + dx, b + dy) - z(a, b)]/dy gives- [z(a + dx, b + dy) - z(a, b + dy)]/dy + [z(a, b + dy) - z(a, b)]/dy = [z(a + dx, b + dy) - z(a, b + dy)].[dx/dy]/dx + [z(a, b + dy) - z(a, b)]/dy = [∂z/∂x]. [dx/dy] + [∂z/∂y] Note that if we take the point (a, b) on some curve in x-y plane, say f(x, y)=f(a, b), then the directional derivative of the function z = z(x, y) in the direction along the curve at point (a, b), which is along the tangent to the curve at point (a, b), will use dy/dx value as that calculated from the equation of the curve, that is f(x, y) = f(a, b). The equation of the curve may be given in parametric form. Or, it may be given in some other coordinate system. Directional derivative with respect to a variable in one coordinate-system in terms of directional derivatives with respect to variables in other coordinate-system can be found easily using the relations among the variables of two systems. Now, we would like to find the direction and magnitude of the maximum rate of change of function z = z(x, y) at point (a, b). Note that at the point (a, b), the value of the function is z(a, b). The set of all points for which the value of the function is same as at point (a, b) will correspond to a curve whose equation will be z(x, y) = z(a, b). Such curve is called equi-valued curve for the function z(x, y). Along this curve at any point, the change in the value of the function will be zero. Therefore, dz/dr = 0, along this curve. That means [∂z/∂x]/ √[1 + (dy/dx)2] + [∂z/∂y].[dy/dx]/ √[1 + (dy/dx)2] will be zero along this curve. Hence, we can find the direction along the curve z(x, y) = z(a, b) at point (a, b) as given by the value dy/dx, which is in fact the slope of tangent to the curve at point (a, b). It comes out to be -[∂z/∂x]/[∂z/∂y]. Now it is clear that if a new value of the function slightly greater than that at point (a, b) is considered, the set of all points in x-y plane for which the function has this value always, will correspond to a curve in close vicinity to the old curve. For a single valued function, these two equi-valued curves will never cut at any point. It is clear now that the maximum rate of change of function z = z(x, y) at point (a, b) will be in the direction perpendicular to the equi-valued curve through point (a, b). Note that for these two close equi-valued curves, value of dz is same for all points, thus to have dz/dr maximum, dr should be minimum. dr is minimum in perpendicular direction. The slope of this perpendicular direction is the slope of the normal to the curve z(x, y) = z(a, b) at point (a, b). It is [∂z/∂y]/[∂z/∂x], as slope of tangent is: -[∂z/∂x]/[∂z/∂y]. Now, the directional derivative in this perpendicular direction, which is obviously maximum of that in any direction, is calculated by substituting [∂z/∂y]/[∂z/∂x] for dy/dx in the expression for directional derivative. After certain simplification, it comes out to be as √[(∂z/∂x)2 + (∂z/∂y)2]. To tell the direction of maximum rate of change in terms of the unit-vector, we see that the slope tanθ for the direction is [∂z/∂y]/[∂z/∂x]. Thus the unit-vector in this θ direction, cosθ î + sinθ ĵ, evaluates to [[∂z/∂x]î + [∂z/∂y]ĵ]/ √[(∂z/∂x)2 + (∂z/∂y)2]. Finally, the vector giving the magnitude and direction for maximum rate of change of function z = z(x, y) at any general point (x, y) comes out to be [∂z/∂x]î + [∂z/∂y]ĵ. As a terminology, it is called gradient vector of scalar function z(x, y).

Thank you for the video! Just to clarify, if ∂ is used to denote a derivative instead of "d" then is it safe to assume that a function definitely has more than one independent variable? Some of the problems I encounter in my Calculus 3 class make it difficult to tell if y is independent of x or if y is a function of x.

@CaffeineEnjoyer_

6 ай бұрын

if it's ∂f/∂x then y is independet variable and if ∂f/∂y then x is independent

Pretty cool.

thanks

Yet, operators presuppose one or the other, regardless dependencies. Gradient, Divergence and Curl all presume partial derivatives and all must be modified when dependencies become muddled, such as inhomogeneous materials where material 'constants' vary with position; requiring tensor and tensor forms of GDC's. Therefore, it's a bit of a misnomer to say a-priori that a partial-derivative means y is not a function of x (to use the video's example); because the partial-operator depends entirely on the function, not the other way 'round. It's more accurate to say that at a problem's beginning, lack-of-interdependency is determined and partial-derivative notation is imposed throughout the forty next pages of derivation merely to remind the student of that initial assumption.

When you saw this d/dx [sin(xe^yzcosy)] You must consider it was Partials derivatives,.If you won't do that,you will see Mr incredible becoming uncanny.

So, total differentiation is implicit?

I didn't know there are 2 xD we just used "d" and did partial xD

愛因斯坦的相對論? 我沒學過...E=MC平方? 啥?

Now I know the difference

If f(x,y) and y(x), then isn't f just a function of x only all along i.e. f(x)? I never understood this notation

Hmm i see

💖👌🏻✨

So you would use one or the other but perhaps never both. And that is because partial derivatives apply to when x and y are independent inputs to f(x, y). This applies to 3D SURFACES! Total derivatives, on the other hand, apply when x and y DEPEND on each other. Therefore, a total derivative is applied to a linear curve traversing across the x-y plane. In such a situation, f(x,y) only has values along the curve, y(x) across the x-y plane. It is therefore, meaningless to talk about taking the total derivative df/dx, for example, for a 3D surface defined by f(x,y) because f potentially has values for all points on the x, y plane.. However, you can take the GRAD of a surface. This is a vector quantity giving you gradients along each of the axis. The gradient along the x axis provides the x component of the vector and the gradient of the surface along the y axis gives you the y component of the vector. Check out this video for a great insight: kzread.info/dash/bejne/mKWlwbeLZN3Olqw.htmlsi=uRu-WX3oXaa-u6Bd

i love you

Hi! Can you upload full lecture series on different topics. It will be useful for JEE exams.

@fufaev-alexander

Жыл бұрын

Sure, I'll do it :)

@Explorelife12

Жыл бұрын

@@fufaev-alexander Thanks. Looking forward to your lecture series.