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 $x$:
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!
