What could be an example of a real life situation for which an implicit function may arouse?
In real life, while plotting a value against the other, wouldn't it be the case that the function would not be implicitly defined?

This relates to the understanding

Why is the function always differentiated with respect to x? Books and tonnes of sites just let us plug in the rule. But, I haven't found a single resource that states why is the implicit function differentiated with respect to x? I don't know if it is just a convention.

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    $\begingroup$ Maybe you just have an ordinary differential equation like $y'(x) = f(y(x))$ and you trace its solution through time with a simulator. It almost certainly won't have a closed-form solution. $\endgroup$ – Mehrdad Mar 7 '17 at 18:37
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    $\begingroup$ How often in "real life" are we plotting one value against another using a known function? $\endgroup$ – David K Mar 8 '17 at 11:19

A vast majority of scientific experiments can be considered to be collecting data that follows some implicit relation. What do I mean? In a typical scientific experiment, you wish to test or investigate an effect, and you have a number of parameters that you think can possibly determine some effect that can be observed via measured quantities. So what you do is to set up a controlled environment where you keep everything else the same and just change the parameters, each time recording the corresponding measurements. A major hallmark of a scientific theory is that it can be verified by reproducing experimental results that it predicts. But actually there is a catch; the point in time cannot be reproduced, and hence all that an experiment can provide are measurements at distinct points in time, and time ends up as a parameter as well. Yet in many models we assume that the physical laws are invariant over time, and hence we can based on repeated experiments deduce some relation between other variables.

Another common way that implicit relations turn up is in dynamic systems at equilibrium. This is because they are often governed by 'balance' equations that constrain the system, which naturally lead to implicit relations between the relevant variables.

For example, consider a titration experiment where we measure the pH of a solution at different points of the titration. Here you might think that given the analyte and titrant, and the initial amount of analyte, the only parameter that is needed to determine the pH of the mixture is the amount $x$ of titrant added. So the curve that we plot will be pH against $x$:

enter image description here

Notice something; the actual titration process corresponds to moving along the curve shown. But we do not actually care how, and indeed repeating the titration shows that the relation between pH and $x$ is independent of how long we wait between each step of adding more titrant. So we have an example of the first kind of implicit relations that arise from ignoring the time parameter.

Furthermore, even in the special case depicted where we assume complete disassociation of the base BOH, we can mathematically prove under standard assumptions (mass balance and charge balance and ionic product of water) that the pH is not a simple function of $x$. It actually turns out to be the logarithm of the positive root of some cubic equation whose coefficients are some rational functions of $x$. So even if we ignore time and just consider $x$ as the independent variable and the pH $p$ as the dependent variable, the relation between $x$ and $p$ can still be considered as an implicit one of the form $a(x)e^{3p}+b(x)e^{2p}+c(x)e^p+d(x)=0$ where $a,b,c,d$ are functions. It so happens that we can algebraically solve for $p$, but it is unimportant. We may also be interested in how fast the pH changes with respect to $x$, in which case it is easy to just use implicit differentiation to obtain the mathematical solution!

  • $\begingroup$ makes more sense. The first answer did also, but, it was as if the answer were left in the middle. One question: we differentiate implicitly, but doesn't the result give us rather an explicit relationship? dy/dx?? $\endgroup$ – bzal Mar 8 '17 at 3:29
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    $\begingroup$ @bzal: Implicit differentiation does not always give you an explicit formula for the gradient. In the above example, differentiating the implicit equation with respect to $x$ you would get $a'(x) e^{3p} + 3 a(x) \frac{dp}{dx} e^{3p}$ for the first term. It turns out that you can then factor $\frac{dp}{dx}$ out from three of the terms, and hence get an expression for it in terms of $p$ and $x$. If you did not know how to algebraically solve the cubic, you would not be able to express it in algebraic form in terms of $x$ alone. In any case using the implicit differentiation is much cleaner. $\endgroup$ – user21820 Mar 8 '17 at 5:50
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    $\begingroup$ @bzal: By the way, you can accept my answer if you like it best. I hope I've convinced you that most real-life situations involve implicit functions because we purposely discard irrelevant parameters or because the phenomena arise from multiple simultaneous constraints. =) $\endgroup$ – user21820 Mar 8 '17 at 5:58
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    $\begingroup$ @bzal: We are free to differentiate in any way we want as long as it makes sense to us. You see, when we plot the titration graph we could very well have plotted $x$ against pH. Nothing was stopping us. But we usually consider observed pH as a consequence of choice of $x$, so we are interested in pH with respect to $x$. Similarly, we are often interested in how fast the pH changes as we add more titrant (BOH), so that's why we differentiate with respect to $x$. $\endgroup$ – user21820 Mar 8 '17 at 13:14
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    $\begingroup$ best answer!! thanks :-) $\endgroup$ – bzal Mar 8 '17 at 13:17

In a "real life situation" you usually have a model that relates the values of two variables. For example, the number $E$ of software engineers looking for jobs and the average salary $S$ offered are related. When the industry is hot companies need developers and pay more to attract them. When schools graduate more engineers than the industry needs, starting salaries are lower.

An economist could ask two kinds of questions:

  • How does an increase in $E$ affect $S$?

  • How does an increase in $S$ affect $E$?

You can answer these questions using a plot with $E$ on one axis and $S$ on the other. If you happen to have a formula that models $S$ as a function $f(E)$ then the derivative of $f$ with respect to $E$ works for the first question. For the second, you need the implicit derivative of $E$ with respect to $S$.

In general, in real life problems you may not have one particular variable that you think of as the independent variable that others depend on. You just have relations among variables.

Here's an example from an answer to one your blender questions (https://blender.stackexchange.com/questions/56715/what-to-pay-attention-when-modelling-for-animation):

You need to consider how the model will be animated, think of the arm being lifted and swung around, think of extremes like swinging from an overhead bar or doing a high roundhouse kick, your model needs to be deformed by the armature without the shoulders or hips getting kinked and looking unnatural.

In this case the variables (inside blender) that describe the position of the shoulder are related to the ones that describe the position of the extended arm. Your animation makes small changes in the arm as a function of small changes in the animation parameter time. Blender figures out how a small change in the arm changes the shoulder. That's essentially implicitly differentiating the function relating shoulder and arm variables. It's implicit because there is no calculus-like formula inside blender to differentiate. The software computes the small change numerically. The calculus analog would be looking at $\Delta(\text{shoulder})/\Delta(\text{arm})$ before you take the limit of that difference quotient. If you wanted to model how the arm changed when you moved the shoulder you would just use the reciprocal of that fraction.

  • $\begingroup$ To add on to this, I'm a data scientist at a large engine company, I get time series data of things like engine speed, and fuel temperature, and am often asked to find relations between the two. I usually use an ANN, which is just a set of matrix operations. That ANN can be (numerically) differentiated to help explain how engine speed and fuel temperature change together $\endgroup$ – Mohammad Athar Mar 7 '17 at 13:53
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    $\begingroup$ @MohammadAthar Yes, Time series data for two quantities $A$ and $B$ often lead to this kind of question. The independent variable (from the point of view of the data) is time, but the relation between $A$ and $B$ is what you care about. You can view the example in my answer this way. $\endgroup$ – Ethan Bolker Mar 7 '17 at 13:56
  • $\begingroup$ @EthanBolker , implicitly defined functions would mean the dependency of a variable, not defined. Does, it mean that it is rather a parametric differentiation? (A parameter that would defined the model that relates the two variables?), If there is not a parameter than what would the model be defined by? $\endgroup$ – bzal Mar 7 '17 at 14:20
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    $\begingroup$ @bzal I don't quite understand the question in your comment. I've edited my example to include another example, where there is an implicit "time": parameter; maybe that helps. If not, I'm afraid I can't help any more. $\endgroup$ – Ethan Bolker Mar 7 '17 at 14:57
  • $\begingroup$ @EthanBolker: Yes indeed in many real-world phenomena, time could be argued to be one of the independent variables, but may not be relevant to the relation at hand. In my answer I describe a concrete example from chemistry. =) $\endgroup$ – user21820 Mar 7 '17 at 17:03

You have a circular curve of which you want to know the diameter. You have a measuring wheel to get the arc length and a meter to get the chord. (This is a true real-life situation in road management.)

Then geometry tells you that

$$\sin\frac ad=\frac cd.$$

Denoting the unknown $y:=a/d$ and the independent variable $x:=c/a$, we get the beautiful implicit equation

$$\sin y=xy.$$

Note: we can express the solution as

$$y=\text{sinc}^{-1}x$$ where $\text{sinc}$ denotes the cardinal sine function, but it's inverse is not accepted as valid for a closed-form expression.

$x$ usually denotes an independent variable and $y$ the dependent one. This is why you are interested in $dy/dx$ rather than in $dx/dy$, even though these two derivatives are just the inverse of each other.

  • $\begingroup$ so, would it mean, that it is a convention after all? But, the fact that functions are defined implicitly would mean that the dependency are not explictly defined, isn't it? $\endgroup$ – bzal Mar 7 '17 at 14:14
  • $\begingroup$ @bzal: hem, your questions aren't very clear. In "it is a convention": what is it ? And what do you mean exactly by "not explicitly defined" ? $\endgroup$ – Yves Daoust Mar 7 '17 at 14:17
  • $\begingroup$ I meant the way how functions are differentiated with respect to x $\endgroup$ – bzal Mar 7 '17 at 14:22
  • $\begingroup$ I was confused at the point, " you are interested...", if the function does not respect the dependency of the variables then why would we be interested in assuming x as the independent variable.(The line "x usually ......", I think implied so) $\endgroup$ – bzal Mar 7 '17 at 14:24
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    $\begingroup$ @bzal: you probably mean that "explicitly defined" means "explicitly defined" then. Ok. $\endgroup$ – Yves Daoust Mar 7 '17 at 15:26

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