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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

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.

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

Basic stuff but excellent explanation!

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

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

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

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

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

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.

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

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

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.

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

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

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 .

...gerade habe ich eine KZfaq-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 < d > 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

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

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

Short and precise. Thank you for this!

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

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.

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).

1:43 very well spent. Thanks very much!

Many students would simplify both to f/x😲

Great stuff, Sir keep it up.

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

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

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

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

awesome video! thank you!

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.

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

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.

Best video explaining the difference. Good job.

nice video!

Straightforward explanation, thank you sir

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

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

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

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

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.

thank you

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

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

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

Thank you very much sir! very useful video

Omg never knew maths was this interesting!

Thank you amazing video

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

If only my calculus teacher had explained it this simply 😑

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.

This is the clearest explanation I've seen

Thank you!

I used to think that the other variable is taken as a constant in case of total derivative too. Hence my belief that partial derivatives are pointless.

thanks bro

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.

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

Gracias chavall!!!

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 information

Short and to the point.

Found helpful 🎉

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

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

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

love it❣

Thank you sir

Hi, I have one question for you : Let us suppose that we have a function f(q,p) such that f(q,p) = (q+p)^N where N is a positive integer and p is a function of q such that p = 1-q. We can easly see that the function f is equal to one and df/dq = df/dp = 0. My question is, if we use the partial derivative del f/del p, we consider the two variable p and q independant so del f/ del p = N(p+q)^(N-1) which is equal to 1 if N>2. That is correct? Thank you.

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.

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

Beautiful explanation.

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.

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

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.

Hmm i see

This is merely a "special" case of the total derivative. For anyone keen to to know what total differentiability really means look up "total derivative" on Wikipedia, for example. This video does not grasp half of the definition of the total derivative. To describe in a few words what the difference is: A function is totally differentiable if there exists a linear map that approximates our function _on all dimensions of the domain_ (for the one dimensional case total and partial differentiability are the same, for higher dimensions that is generally not the case). a function is partial differentiable if you can find for each dimension of the domain of your function a linear approximation. If your function is partial differentiable in every direction (each single dimension, that means you find for every dimension/"variable" a one-dimensional linear approximation) AND all those partial derivatives are continous, THEN and only then your function is also totally differentiable.

wow! nice video

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

u r king

Very nice

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

作為教學頻道傳遞錯誤的知識真的不行。 這部影片完全沒有解釋到什麼是全微分，而是拿方向微分來偷換概念。 討論全微分的時候，df/dx這個符號的x不應該被理解成x-y平面(定義域)的第一個變量，而應該是x-y平面(定義域)裡的一個向量。 留言區一大堆人被誤導，我很難過。 ========== As a knowledge popular channel, you should NOT spread misleading messages. This video has mentioned NO content of total derivative but directional derivative. For total derivative, you should not take the symbol "x" of "df/dx" as the first component of x-y plane (or the domain of f). Instead, it represents a VECTOR of x-y plane (or the domain of f). A lot of people have been misled. That's not cool.

Sir how does it become 6x in differentation 0f a/ax*3x²

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.

Is there a limits equivalent for partial derivatives?

Does this makes a difference in calculation?

thanks

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

That's really helpful!

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

Fantastic

Sir 49% helpful 😅

The symbol for partial differentiation is not a curved del, but a stylized cursive d.

So nice thanks

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: kzfaq.info/get/bejne/nNehkriDmeDMln0.htmlsi=uRu-WX3oXaa-u6Bd

I don't like that the use of one symbology over another indicates whether Y is an expression that depends on X or not. It's needlessly confusing. It adds yet another thing that you just have to know beforehand, instead of just simply stating whether y depens on x or not.