# Talk:Partial derivative

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One of the 500 most frequently viewed mathematics articles.

## Thanks!!!

Thanks to all the writers of this page! Saved my behind (-:

Thanks again, Sam Krupa

Agreed. I've never known what a partial derivate is so I decieded to check it up. Clearly and well written.

--- —Preceding unsigned comment added by 88.114.90.71 (talk) 09:43, 18 March 2009 (UTC)

Yeah, this is really an impressive piece of work. The introduction is really accessible, contrary to loads of the maths articles on wikipedia. My 2 Cents' Worth (talk) 10:42, 3 June 2010 (UTC)

i came to talk to say same things- best intro  in any wilpedia item on math  — Preceding unsigned comment added by 174.24.33.155 (talk) 19:52, 25 December 2015 (UTC)


## Partial derivative of Area of a circle

Since there is only one variable in the formula for the area of a circle, as the article mentions, isn't the example using it a bad one? (Unsigned comment by 68.78.139.53 on 17 May 2006)

Although it's been many, many years since I've done any maths, I would agree - this example does seem pointless. --A bit iffy 09:54, 11 June 2006 (UTC)
I agree. I've removed it. --Spoon! 02:22, 11 September 2006 (UTC)

## Disagree with recent changes

The third and the fourth paragraph, that is the discussion about the total derivative and the Jacobian are way out of place. Those things are inserted in the middle of the discussion about the partial derivatives, and that is inappropriate.

All that stuff needs to go in the last section, where the gradient is discussed. Other opinions? Oleg Alexandrov 02:27, 23 Mar 2005 (UTC)

## Comparison with "d" notation

I'd like to see a section addressing the semantic difference between ∂ and d. For example, if y is only a function of x, then what is the difference between ∂y/∂x and dy/dx. Also, in sloppy engineering notation I've seen dy appear alone; I think I've seen ∂y alone as well. This may be rigerous use of Infinitesimals or it may be engineering shorthand; I'm not sure. —BenFrantzDale 02:00, 5 December 2005 (UTC)

I would agree with such a section, but preferably at the bottom of the article; otherwise I would think that it may confuse more than illuminate. Some connections with the Leibniz notation may be made. Oleg Alexandrov (talk) 02:59, 5 December 2005 (UTC)
Now I see, the page I needed to look at was total derivative. —BenFrantzDale 17:40, 15 December 2005 (UTC)
What actually is the difference between dy/dx and δy/δx? Is it just preference? My 2 Cents' Worth (talk) 10:44, 3 June 2010 (UTC)

## Vote for new external link

Here is my site with partial derivative example problems. Someone please put this link in the external links section if you think it's helpful and relevant. Tbsmith

http://www.exampleproblems.com/wiki/index.php/PDE:Integration_and_Seperation_of_variables

## Ordering of the sections

I am a bit perplexed as to why the formal definition is at the end and the example is at the beginning, I think some reordering is in order here! Retardo 22:13, 5 May 2006 (UTC)

## Surfaces?

In computational mechanics, it is common to denote the surface of a volume as S=∂V. My sense is that this has some underlying meaning in differential geometry. Is it just a notational convenience or is there some rigerous extension of partial differentiation onto volumes? Also, how does this relate to the dV you find in a volume integral? (There dV basically means "one tiny point in V" whereas ∂V basically means "one thin sliver of the surface"; does this relate to the difference between ∂ and d?) —Ben FrantzDale 23:29, 18 May 2006 (UTC)

In topology, ∂S denotes the boundary of a surface S. As far as I know, this is unrelated to the symbol's use to denote a partial differential (I've only just started seeing this formally, so I may be wrong). I also read this on an article here on Wikipedia which was about the different uses of the symbol; an article that I can't seem to locate now. If anyone can find it, I'd like to know. Anyway, I think this makes sense since the boundary of a volume is a surface, as the boundary of an area is a path (at least intuitively). Hope this helps. Commander Nemet 23:25, 31 May 2006 (UTC)

## Integration equivalent of partial derivative

Since ${\displaystyle F'(P)={\frac {dF(P)}{dP}}\,\!}$ and (with respect to P) ${\displaystyle F'(P,Q)={\frac {\partial F(P,Q)}{\partial P}}\,\!}$, doesn't it follow that ${\displaystyle F(P_{d},Q)-F(P_{b},Q)=\int _{P_{b}}^{P_{d}}F(P,Q)\partial P\,\!}$?  ~Kaimbridge~18:44, 20 May 2006 (UTC)

Yes, that is true, assuming that by the integrand you really meant ${\displaystyle F_{P}(P,Q)}$ instead of F itself. Also, I believe that the differential here would be just dP, not ${\displaystyle \partial P}$, though I can't give you a good reason why. What you would get after this integration is a function of Q, which, if you were doing a double integral, would be integrated in the next step. I always thought of the double (or triple, etc.) integral as the "equivalent" of the partial derivative. Commander Nemet 03:09, 1 June 2006 (UTC)
Actually, I meant ${\displaystyle F'(P,Q)\,\!}$. In terms of full notation, should it be
${\displaystyle F(P_{d},Q)-F(P_{b},Q)=\int _{P_{b}}^{P_{d}}F'(P,Q)\partial P=\int _{P_{b}}^{P_{d}}G_{p}(P,Q)\partial P\,\!}$, or should ${\displaystyle F'(P,Q)\,\!}$ be ${\displaystyle F'_{p}(P,Q)\,\!}$, with a subscript, too? As someone asked above in "Comparison with "d" notation", if you have "dP", you know it's regarding P, not Q, so what does ${\displaystyle \partial P\,\!}$ add? You could just as easily say
${\displaystyle E'(P)=F'(P,Q)=\sin(P)\cos(Q)\,\!}$, in which case
${\displaystyle E'(P)dP=F'(P,Q)\partial P\,\!}$, right? In terms of it being a "double integral", a double integral integrates one variable, then the other. In the example I give above, only "'P'" is integrated.  ~Kaimbridge~19:28, 4 June 2006 (UTC)
In regards to your first question, the notation that I learned for partial derivatives includes, in this case, ${\displaystyle {\begin{matrix}{\frac {\partial F(P,Q)}{\partial P}}\end{matrix}}}$, ${\displaystyle F_{P}(P,Q)}$, and ${\displaystyle \partial _{P}F(P,Q)}$. ${\displaystyle F'(P,Q)}$ is not used because it does not indicate which variable the derivative is taken with respect to.
In regards to your second question, I'm not really sure what you're asking, but I can tell you that, as far as I know, the partial differential symbol is used only in derivatives, so in the integral, for example, it would be just dP. I think the reason for this is that a differential is only "partial" in the context of a derivative. To me, at least, having a differential be "partial" otherwise would be meaningless. Commander Nemet 22:49, 5 June 2006 (UTC)

## Greek or Cyrillic?

It seems that 68.103.26.177 has changed the introduction to say the symbol is from the Greek instead of Cyrillic alphabet. I'm trying not to be hasty and call it vandalism, but unless there is some deeper meaning I am unaware of, the partial differential symbol looks a heck of a lot more like the cursive de (Cyrillic) than a Greek delta. Furthermore, the intro now purports to say that the Greek letter in question is actually a "d"—something that doesn't exist in that alphabet. Commander Nemet 23:25, 31 May 2006 (UTC)

• I agree with you that this is just wrong, if not vandalism. I therefore reverted it to the correct version. PanchoS 00:29, 29 June 2006 (UTC)
• I think the firs person to use that notation was leibniz
According to Cajori's History of Methematical Notations v. 2, p. 220 Leibniz used a 'δ' for the partial derivative. Thomasmeeks 17:31, 11 December 2006 (UTC)

## Can we get a computerized notation set up?

Many other mathematics-related articles have an implementation of the topic in various programming languages. Will anybody set up this page with such an example? -- kanzure 16:41, 19 July 2006 (UTC)

## Is modern origin of ∂ Cyrillic?

If so, please cite here. I'm doubtful based on # of hits for Google of:

• "partial derivative" Greek
• "partial derivative" Cyrillic

Earlier notation used Greek small delta δ, which looks like than the Cyrillic Б. If no one can give an authoritative cite for Cyrillic, my vote would be for deleting sentence so suggesting it. Greek origin looks more plausible. Thomasmeeks 02:25, 28 November 2006 (UTC)

The answer to the question in this section title seems to be in the following classic article:
Florian Cajori, "The History of Notations of the Calculus." Annals of Mathematics, 2nd Ser., Vol. 25, No. 1 (Sep., 1923), pp. 1-46
accessible through JSTOR from many colleges and universities. Thomasmeeks 01:08, 29 November 2006 (UTC)
Should the reference to the Cyrillic letter in the lead be deleted? For now, I believe so. Florian Cajori's A History of Mathematical Notations (1928-29), still widely regarded as definitive for the period through its writing, makes no reference to the cursive Cyrillic letter de in the section on partial derivatives (v. 2 pp. 220-42). He refers only to the "rounded letter" after referencing earlier partial notations, including the 'ό' and 'd'. No other scholarly sources in the math reference section or the relevant sections of book collections on math history at a good university library make the Cyrillic connection. Nor does a search of Google Scholar give any primary source for: Cyrillic "partial derivative." So, I don't believe that the Cyrillic reference meets the Wikipedia: verifiability requirement. Thomasmeeks 18:50, 11 December 2006 (UTC) ('rounded' for 'curved' edit Thomasmeeks 20:16, 11 December 2006 (UTC))Thomasmeeks 20:18, 12 December 2006 (UTC)

Is there a WP policy that discourages arguments based on the number of Google 'hits'? If not, can someone please propose one? Here for example: From searching Google today:

• "partial derivative" American ~54000
• "partial derivative" English ~36400
• "partial derivative" French ~24000
• "partial derivative" Greek ~12300
• "partial derivative" Cyrillic ~519

On this basis we'd conclude that: Americans invented partial derivatives, and they probably spoke (a form of) English; their invention was aided by the French and quickly adopted by the Greek; and Russians generally know nothing about partial derivatives.
But let's also search >"partial derivative" Amrica<. It returns ~35800 'hits'. At least, Google claims that number of hits. It can't be verified. Disagree? Okay, then tell me what hit number 10000 is.
And what about other search engines???
—DIV (128.250.204.118 01:34, 15 November 2007 (UTC))

## Deletion of 2nd sentence of 2nd paragraph

Deletion from the previous Edit of the article of:

'∂' also corresponds to the small Greek delta 'ό' which was also used before the 20th century as partial derivative notation.

The problem is that that sentence follows a related parenthetical comment. In retrospect (I introduced the second sentence), if the sentence is going to be included at all, there is a case that both should be parenthetical (to avoid breaking the substanive exposition) or neither (possibly as a footnote or short separate section on the origin of the notation). Thomasmeeks 14:56, 29 November 2006 (UTC)

## a step backwards

"a partial derivative of a function of several variables is its derivative with respect to one of those variables with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary"

the total derivative is, in fact, the sum of the partial derivatives. i seem to remember explaining this before. but no matter. keep removing other people's contributions if you dont understand them —Preceding unsigned comment added by 212.159.75.167 (talk) 20:26, 13 October 2007 (UTC)

I disagree, the total derivative is not the sum of the partials. It is possible for all the partial derivatives to exist and yet the function not even be continuous (let alone have a total derivative). This is mentioned in the section *Formal Definition* : "even if all partial derivatives ∂f/∂ai(a) exist at a given point a, the function need not be continuous there." Brian Maurizi 72.51.124.202 (talk) 19:20, 21 April 2010 (UTC)

## numericalization

We should also discuss the numericalization in this article. At least illustrate how to numericalize the 2nd derivative. anyone agree? Jackzhp (talk) 15:22, 3 May 2008 (UTC)

## Need Help

Hi, i'm currently studying on this topic(partial derivative) and i need some help to solve these problems.

1. The variable x and r are defined as

x = r cos θ and y = r sin θ.

a) Explain what is meant by i)( ∂y/∂r)θ and ii)( ∂y/∂r)x . b) Determine the values of (i). c) Determine value of (ii) by eliminating θ in the expressions for x and y. d) Determine the value of (ii) by the method of function of a function. e) Determine the value of :( ∂y/∂r)θ .( ∂y/∂r)x using method of b) and c).

2. If u = ax +by and v = bx – ay, determine the value of:

               (∂u/∂x)y.( ∂x/∂u)v


3. A, B and C are the angles of a triangle and u is a function defined as

                     u =  sin A sin B sin C.


i) Express u in terms of A and B only. ii) Prove that u is a maximum if the triangle is equilateral.

4. A vessel to be constructed is in the shape of a cylinder of radius r with equal conical ends, the semi-vertical angle of each cone being α. If V is the volume and S is the total surface area, show that

            S = 2V/r  + 2πr²(cosecα  - 2/3cotα )


If both r and α can vary, show that for a vessel of fixed volume and minimum surface area, cos α = 2/3, and determine r in terms of V.

Is there anyone able to enlighten me on this problems? THANKS a million... —Preceding unsigned comment added by Xpsky (talkcontribs) 05:35, 2 February 2009 (UTC)

Please ask this question at Wikipedia:Reference desk/Mathematics. You will most certainly get a response, if you further add your input to the answering of these questions. In general, the talk page is not the most appropriate place to post questions. Further, the problems you quote are rather simple in nature, and similar worked examples can be found in a basic textbook on multivariable calculus. --PST 04:38, 22 May 2009 (UTC)

## verify total derivatives

Considering total derivatives article, I feel that the equations should be

${\displaystyle {\frac {\operatorname {d} V}{\operatorname {d} r}}=\overbrace {\frac {2\pi rh}{3}} ^{\frac {\partial V}{\partial r}}+\overbrace {\frac {\pi r^{2}}{3}} ^{\frac {\partial V}{\partial h}}{\frac {dh}{dr}}}$
${\displaystyle {\frac {\operatorname {d} V}{\operatorname {d} h}}=\overbrace {\frac {\pi r^{2}}{3}} ^{\frac {\partial V}{\partial h}}+\overbrace {\frac {2\pi rh}{3}} ^{\frac {\partial V}{\partial r}}{\frac {dr}{dh}}}$
${\displaystyle k={\frac {h}{r}}={\frac {dh}{dr}}}$

${\displaystyle {\frac {\operatorname {d} V}{\operatorname {d} r}}=\overbrace {\frac {2\pi rh}{3}} ^{\frac {\partial V}{\partial r}}+\overbrace {\frac {\pi r^{2}}{3}} ^{\frac {\partial V}{\partial h}}{\frac {\partial h}{\partial r}}}$
${\displaystyle {\frac {\operatorname {d} V}{\operatorname {d} h}}=\overbrace {\frac {\pi r^{2}}{3}} ^{\frac {\partial V}{\partial h}}+\overbrace {\frac {2\pi rh}{3}} ^{\frac {\partial V}{\partial r}}{\frac {\partial r}{\partial h}}}$
${\displaystyle k={\frac {h}{r}}={\frac {\partial h}{\partial r}}}$

Please verify this. —Preceding unsigned comment added by Landroni (talkcontribs) 15:00, 18 February 2009 (UTC)

Yes, you're right. I have corrected it, and removed the attention box. RupertMillard (Talk) 10:04, 25 February 2009 (UTC)

## Three arguments to a two-variable function?

The Introduction chapter mentions "to find the tangent line of the above function at (1, 1, 3)", as well as other instances of "(1, 1, 3)". But the function in question is f(x,y), which takes only two arguments. Its solution at (1, 1) is 3, though, as well as ƒ(1, 1) = 3. Perhaps it is an error that has slipped by?

The question above was first posed by User:82.130.38.44, on 23:35, 19 April 2009 (UTC) but later removed by him. Perhaps did (s)he find a good explanation that (s)he can give to everybody? At this moment, I rather think (s)he was right to suspect an error. --Bdmy (talk) 09:17, 20 April 2009 (UTC)
As 82.130.38.44 determined, ƒ(1, 1) = 3, so (1,1,3) are the (x,y,z) coordinates of a point on the surface. This is a good habit, as some functions will have multiple values of z for some x&y, in which case the derivatives at these different points will most probably be different. RupertMillard (Talk) 09:57, 20 April 2009 (UTC)
Of course (1,1,3) is the point on the surface, but is it really a good way of phrasing things? You wouldn't talk of the derivative of the exponential function ${\displaystyle e^{x}}$ at the point (1, e). Including the value as additional variable does not make any reasonable sense in my silly example, and I am not sure it does in this article (especially without a warning – for me it is just confusing). Also I don't agree with your point that some functions have multiple values at (xy). For me they are just not functions in that case. You probably think about surfaces defined by an implicit equation, and that can be described by several "charts", involving several functions. --Bdmy (talk) 10:25, 20 April 2009 (UTC)

This article discusses intuition but fails to discuss the main properties of the partial derivative; only at the end is the formal definition stated. For example, many facts found in a basic calculus textbook are not discussed, regarding the partial derivatives. What worries me, is that even the content of a basic calculus textbook is not adequate enough for this concept. --PST 04:35, 22 May 2009 (UTC)

## Question

Doesn't f sub x = the partial of f / the partial of x? (not the partial of f / the partial of y)

see basic definition, where f sub a takes y to be the variable.

No, that's not what it means in that case. Read the defitions given just before that. The subscript doesn't represent any sort of derivative at all. That conflicts with the notation used elsewhere in the article, so that's potentially confusing. But the paragraph preceding that part is explicit about what the notation means. Michael Hardy (talk) 22:36, 20 September 2009 (UTC)

## Good Job!!

My compliments to whoever has guided this wiki page, be it a group or an individual.

The introduction is EXCELLENT! — Preceding unsigned comment added by 220.165.68.56 (talk) 07:13, 8 January 2013 (UTC)

## Inconsistency regarding Second-order mixed derivatives

In the Notation-section it says:

${\displaystyle {\frac {\partial ^{2}f}{\partial y\,\partial x}}=...=f_{xy}}$

But in the Higher order partial derivatives-section it says:

${\displaystyle {\frac {\partial ^{2}f}{\partial x\partial y}}\equiv ...\equiv f_{xy}}$

So my basic question is the meaning of ${\displaystyle {\frac {\partial ^{2}f}{\partial x\,\partial y}}}$:

1. Does it mean, that you first differentiate with respect to x and then with respect to y, because that is the order of the denominator?
2. Or does it mean, that you first differentiate with respect to y and then with respect to x, because derivative operators always work on the stuff, that comes after it? (So basically read from right to left, rather from left to right?) 134.60.31.73 (talk) 16:27, 10 April 2015 (UTC)
The theorem on symmetry of second derivatives says that the two are equal--you get the same thing regardless of the order in which you take the derivatives. (This is mentioned at the end of the subsection "Formal definition".) So it doesn't matter how you interpret the notation, though it is standard to mean ${\displaystyle {\frac {\partial ^{2}f}{\partial x\partial y}}\equiv \partial \left({\frac {\partial f}{\partial x}}\right)/\partial y,}$ as is implicit in the H matrix in the article Hessian matrix. Loraof (talk) 20:09, 9 May 2015 (UTC)
Thank you for your answer. But in the same article symmetry of second derivatives is says that this symmetry does only hold under special requirements, so it is not generally true, see here [[1]]. So in the general case it does matter how I interpret the notation. Also in your mentioned article Hessian matrix I cannot find any passage that supports your mentioned "standard", perhaps you can elaborate? I would have chosen the opposite interpretation (my second alternative from my initial post).134.60.31.73 (talk) 12:06, 19 June 2015 (UTC)

## Assessment comment

The comment(s) below were originally left at Talk:Partial derivative/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

 Needs references. Geometry guy 01:21, 23 June 2007 (UTC) Doesn't f sub x = the partial of f / the partial of x? (not the partial of f / the partial of y) see basic definition, where f sub a takes y to be the variable.

Last edited at 20:47, 20 September 2009 (UTC). Substituted at 02:27, 5 May 2016 (UTC)

## f´(x)=f(x)?

I think the red term in the quote

The partial derivative of a function ${\displaystyle f(x,y,\dots )}$ with respect to the variable ${\displaystyle x}$ is variously denoted by ${\displaystyle f_{x}^{\prime },\ {\color {red}{f_{x}}},\ \partial _{x}f,\ D_{x}f,\ D_{1}f,...}$

can not be correct, can it? I think that would be the undifferentiated function, not the derivative. --Yukterez (talk) 02:01, 4 September 2017 (UTC)

## Use of this notion in the context of math-based sciences or applied math

It seems that there are non-trivial aspects of the use of partial derivatives in the context of math-based sciences aka applied mathematics. An example is one connected to the Gibbs-Duhem equation. I also notice some objection to the expression math-based sciences recently removed by user:Deacon Vorbis.--82.137.15.34 (talk) 11:35, 25 October 2017 (UTC)

You haven't asked a question or made a proposal or anything, but I'll add that essentially all science is "math-based". --Deacon Vorbis (talk) 14:31, 25 October 2017 (UTC)
True (almost), but there are some areas from particular sciences which are more qualitative-descriptive.--82.137.11.191 (talk) 21:05, 25 October 2017 (UTC)
A very interesting area or (applied) science (that is) rather qualitative (math is not required, at least as I know, in the educational system preparing those professionals) is medicine! (Thoughts?)--82.137.15.37 (talk) 00:12, 26 October 2017 (UTC)
IP, i think the issue with your addition is that there are many, many applications in the sciences. In medical R&D partial derivatives are important in modelling blood flow (imp for cardiology, medical device development), pharmacology, medical imaging, nuclear medicine. But not doctors so much. Radiologists, when determining what kind of scan to give someone maybe, but they mostly have software that does this for them. Jytdog (talk) 01:30, 26 October 2017 (UTC)
Very intriguing and/or paradoxical the situation you mention about applications of math in medical and physiological context, but not directly influencing the formation/education of medical professionals involved in formulating medical diagnoses. Perhaps this has to do with a misconception of a historical nature of bad legacy having its origin in Medieval times (before the Scientific Revolution) when math was not very appreciated or developed in those medieval times when only the literary and rethorical aspects from the Classics thinkers of Antiquity were highly esteemed and medicine was seen as craft or practical art in contrast with the liberal arts.--82.137.11.35 (talk) 11:40, 26 October 2017 (UTC)
It also seems that medical doctors feel a less stringent necessity to use and develop abilities/skills involving mathematical thinking and mathematical modelling applied in their activities. The mentioned applications with medical impact that use mathematical notions are therefore left to be developed by other categories of professionals who have another conceptual perspective and a better appreciation of the importance of some math notions such as engineers. After all, the above mentioned applications can be envisioned to belong more in the biomedical engineering perspective.--82.137.11.35 (talk) 11:53, 26 October 2017 (UTC)

### Example of use in thermodynamics

The expression for the extensive Gibbs free energy G of a mixture is a sum of weighted partial derivatives, weights being the molar amounts of the components ni.

${\displaystyle G=\sum _{i=1}^{m}n_{i}{\bar {G_{i}}},}$

where ${\displaystyle {\bar {G_{i}}}}$ is the partial molar ${\displaystyle G}$ of component ${\displaystyle i}$ defined as:

${\displaystyle {\bar {G_{i}}}=\left({\frac {\partial G}{\partial n_{i}}}\right)_{T,P,n_{j\neq i},({\frac {x_{1}}{x_{3}}})}}$

The first additional relation can be expressed equivalently in term of total molar Gibbs energy of the mixture ${\displaystyle {\tilde {G}}}$ using mole fractions xi instead of ni where xi is obtained by diving each weight (called mole fractions) by the sum of all ni in the system and using the property that the sum of xi equals 1.

${\displaystyle {\tilde {G}}=\sum _{i=1}^{m}x_{i}{\bar {G_{i}}},}$

The weights xi can be represented geometrically/graphically in case of ternary and quaternary systems using ternary plot and quaternary plot in equilateral triangle and tetrahedron, etc.)--82.137.9.233 (talk) 15:45, 26 October 2017 (UTC)

The last relation can have the derivative taken at with respect to one of the weights, say x2 at a constant ratio of the other 2 weights x1 and x3 because when the variable x2 varies, only the sum of the other mole fractions can vary:--82.137.9.233 (talk) 16:07, 26 October 2017 (UTC)

${\displaystyle {\tilde {G}}=\sum _{i=1}^{m}x_{i}{\bar {G_{i}}},}$

From the sum of mole fractions equals 1 follows that:

${\displaystyle dx_{3}=-(dx_{1}+dx_{2})}$
${\displaystyle \left({\frac {\partial x_{1}}{\partial x_{2}}}\right)_{\frac {x_{1}}{x_{3}}}=-{\frac {x_{1}}{1-x_{2}}}}$--82.137.12.199 (talk) 00:25, 27 October 2017 (UTC)