Real life situation for an implicit function 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.

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.
