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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.
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.
@@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.
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.
@@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.
@@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!
@@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
@@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.
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.
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.
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.
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.
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.
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.
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.
@@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.
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.
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.
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.
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 .
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: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-fqq_UR4zhfI.htmlsi=uRu-WX3oXaa-u6Bd
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.
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).
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.
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
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.
Me too. My teacher in Calc3 said that since you write down the variable you are differentiating with respect to then it doesn't really make a difference on the notation. I assume he said that because we were always given the functions. Plus later we used the chain rule with which it was necessary to understand which function depends on which variable... Still though it wouldn't matter much on the notation idrk... Nevertheless this video did explain it quite simply.
On another note, in physics the teachers never even bothered using the partial derivative notation and they always use the d/dx symbol... They also say that it doesn't make a difference since you are writing down the variable you are differentiating with respect to...
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).
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.
That's because a better way to teach and understand the difference, is that by taking a partial derivative, you intentionally hold all other variables fixed while you take the derivative. It can then be implied that the other variables must not depend on the variable one is taking the derivative with respect to (ie. what he has tried to explain). A better and more intuitive way to think about partial derivatives, is exactly what they are: they are the change in one variable, in the direction wherein all other variables are held constant. This therefore means that y could certainly depend on x, but in the direction such that y is held constant, dy(x)/dx = 0.
So the partial derivative is just the total derivative where dy/dx is known/assumed to be 0? Still doesn't seem like much reason to use completely different symbols.
Perhaps the point is that the partial derivative assumes (without justification) that dy/dx is 0. When that assumption is correct the total derivative is equal to the partial derivative but in more general cases the total derivative is distinct from the partial derivative. I think what we are lacking here is justification for the usefulness of the partial derivitive when the assumptions I not true.
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.
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.
This is straight up wrong info. Usually, the total derivative refers Df, a function that takes in a vector and output the linear transformation that best approximates f around that vector. d/dx is not called a Total Derivative, its just the conventional one dimensional derivative.
@@vishithvas5626 You can just read the Wikipedia article on total derivative: en.wikipedia.org/wiki/Total_derivative EDIT: Upon further reading, apparently physicists just have strange names for things, and what they call a "total derivative" is not the same as what mathematicians call a "total derivative".
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.
There’s some overlap in names here. The upside down capital delta symbol ∇ is often called “del” (as well as “nabla” or apparently “atled”) but the cursive d symbol ∂ is also often called “del” (as well as other names. I’m not really sure why; I’m guessing maybe since it looks kind of like a lowercase delta δ but with part cut out? Even though it’s origins are more directly related to the Latin d).
No glassgubbenen, it's just a good old d (although, in this video it is written in italic (not cursive)) written how they were written before they became upright. The letter ð is just an old d with a bar. d's are still written that way in e.g. fraktur.
@@mattias3668, true. Ironically I knew that, but my lazy attempt at fact checking only consisted of reading Wikipedia and I wrote my comment accordingly. The moral of this story is that bad fact checking might be worse than no fact checking at all.
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).
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.
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.
It's better to think of partial derivatives as the base concept, since this is what the lim_{h -> 0} notation gives us. Indeed, as far as I can tell, you can always just use partial derivatives, and avoid total derivatives entirely. For example, instead of computing (d/dx)(x^2 y^3) you can compute the partial derivative of (x^2 f(x)^3) with respect to x. To answer your question, you can apply implicit differentiation to any equation involving differentiable functions that you're interested in, HOWEVER the result of doing so is a bit vague. This is because to implicitly differentiate an equation, you need to explain to the reader which symbols represent constants and which represent variables. Unfortunately, the underlying foundations of math that are currently in use don't provide any way of expressing this distinction, so if you want to be precise within currently accepted math, you should move to function notation instead. So instead of implicitly differentiating x^2/a^2 + y^2/b^2 = 1, you would instead implicitly differentiate x^2/a^2 + f(x)^2/b^2 = 1. Notice that I've replaced y with f(x), while meanwhile I left 'a' and 'b' alone. This is the currently accepted way of expressing that y is a variable with respect to x; you literally write it as a function of x.
The difference between partial derivative and total derivative is misleading the viewer into thinking you actually applied the derivative to the same function. The letters are just empty symbols. The first derivative was the directional derivative of f along (1,0). The second derivative was the derivative of x -> f(x, y(x)) and chain rule.
It's a very physics-y way of thinking and is sloppy notation (to put down the symbol y instead of y(x)) from a more computer science-based perspective. There's one differentiation operator. You just need to be clear about the inputs and outputs.
So differentiation means to know how the value of function changes as the input changes. Partial differentiation of a function if taken only x means how the function changes as value of x changes and total differentiation means how a function changes as all of the units change
The distinction between total and partial derivative is more an interpretational than a mathematical one. Mathematically, there is usually no such thing as the "total derivative of a function" with respect to one variable. If you have a function in n variables, there is a (partial) deriviative with respect to each of the arguments, i.e. n partial derivatives. A partial derivative is the derivative of a function with respect to one of its arguments. This is a purely mathematical concept. You change one of the arguments of the function and observe how the functional value changes. On the contrary, the total derivative is the derivative of some quantity with respect to some other quantitiy, describing how the first quantity (in reality or some model of reality) reacts to changes of the second. If the relationship between the first quantity and the second can be described by a mathematical formula in the form "first quantity = [some expression in the second quantity]", the derivative of the right-hand side of this formula is called the "total" derivative of the first quantity with respect to the second. Now if the function you are considering is exactly the right-hand side of this equation, total and partial derivative coincide, if the function is not, they do not.
Still doesn't explain what a partial derivative is or is used for - you just show how to differentiate them differently - not really the difference so much between them. Misleading video imo.
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.
Ah yes, another integral part of understanding math ignored and skipped by the teachers and institutions whose sole goal is to teach you these things. Gotta love the education system, am i right?
Good question! The answer to your question is the reason why the chain rule exists. If you have a function f(g(x)), then its derivative is f'g(x)*g'(x) In Leibnitz notation: If y = f(u), and u = g(x), then dy/dx = (df/du) * (du/dx) You can see from the equations for the chain rule that y is a function that depends on u, but u is a function that depends on x. Then doesn't that mean that y is a function that ultimately depends on x? Why yes, yes it is! If this is the case, then why does the chain rule exist? Why do we need the "intermediate variable" (u) if u is a function of x as well? The answer is because sometimes, it is easier to express an equation in terms of an intermediate variable, rather than writing out the equation in terms of its most basic variables. In other words, the relationship between y and x can be expressed much more easily if this relationship is written in terms of the intermediate variable, u rather than directly writing a relationship between y and x. Likewise, the total derivative is, in fact, an application of the chain rule, and so the answer to your question is: yes -- f is a function that depends on x. We take y to be an "intermediate variable" because it makes the math a little more simple. (To be more accurate, the chain rule is a special case of the total derivative, rather than the other way around, but for the purposes of this question/video/comment, it's safe to say that the total derivative makes use of the chain rule.) I hope this helps :)
@@nashs.4206 Thanks for the answer. It makes sense, but I still find it strange. If f(x,y) = x^3 + y with y(x) = x^2, the partial derivative of x would be 3x^2. You can write f(x,y) as xy+y though making its partial derivative y, or x^2. That doesn't seem legal.
@@rikkardo9359 My previous post was about total derivatives, and how it is an application of the chain rule. When it comes to partial derivatives, the important thing to understand is that partial derivatives work under the ASSUMPTION that the variables of interest are totally independent of each other. In your example, you can clearly see that the variables x and y are NOT independent, and so you can't take the partial derivative. You can only take the TOTAL derivaive. I will show you that the derivative of your example function will be the same, regardless of whether you use partial derivatives (which you technically can't because there's clearly a relationship between y and x; but if you absolutely insist on using partial derivatial derivative notation), or total derivatives: Method 1: Substitution f(x, y) = x^3 + y; y(x) = x^2 Substitute y(x) into the expression for f(x,y): f(x, y) = g(x) = x^3 + x^2 dg/dx = 3x^2 + 2x =================================== Method 2: Total derivative f(x, y) = x^3 + y; y(x) = x^2 The equation for the total derivative is: df/dx = (∂f/∂x)*(dx/dx) + (∂f/∂y)*(dy/dx) We can simplify this to df/dx = (∂f/∂x) + (∂f/∂y)*(dy/dx) since dx/dx = 1 Now, applying this equation to your function: ∂f/∂x = 3x^2 + 0 = 3x^2 ∂f/∂y = 0 + 1 = 1 dy/dx = 2x Therefore, df/dx = (3x^2) + (1)*(2x) = 3x^2 + 2x =================================== Method 3: Partial derivatives (which technically isn't possible because the variables are not actually independent) f(x, y) = x*y + y; y(x) = x^2 Note that in this case, I am using the function f(x, y) = x*y + y, instead of f(x, y) = x^3 + y We must use the product rule to take the partial derivative: ∂f/∂x = (x*(∂y/∂x) + y*(∂x/∂x)) + ∂y/∂x Since y = x^2, we have: ∂f/∂x = (x*(2x) + y*(1)) + 2x ∂f/∂x = (2x^2+ x^2) + 2x = 3x^2 + 2x The reason why ∂y/∂x is not 0 is because there IS a relationship between y and x, which is given by y = x^2. As such, the partial derivative of y with respect to x does not vanish. If the two variables were independent, THEN ∂y/∂x = 0. I hope this clarifies some more things! :) EDIT: Now, you might be wondering why it is the case that in method 2, when calculating ∂f/∂x we didn't have to use the product rule, but then in method 3, when calculating ∂f/∂x, we had to use the product rule. Well, the answer should be obvious: in method 3, the function is f(x, y) = x*y + y (which is a product of x and y), whereas in method 2, the function is f(x, y) = x^3 + y
@@rikkardo9359When talking about partial derivatives, you have to assume there is no relation between each variable, so you cannot impose the condition y=x^2, or otherwise you will not be talking about the same function f(x,y)=x^3+y, you would be composing f with the function g(x)=(x,x^2), and dealing with f o g(x)= x^3+x^2 instead.
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
@@undagame7800 This is only the definition of f(x,y). You don't know anything about y. The equation is still exactly as true if y=3 as if it's y=(x^2+9x+7)e^x. Or it could be y=2z or something. We don't know.
