The question is, why implicit differentiation applies to equations [ Because they're not functions ]
As we know, we can apply derivation to the functions, and I know that in implicit differentiation we consider $y$ as a function of $x$, but what that function is?
I had also asked another question which didn't get a satisfying answer for it, it is also related to this matter.
Please, please, please to infinity , don't give Wikipedia link of the theorems.I just wonder about those people who copy/paste the Wikipedia links, if Wikipedia was satisfying, I would have never asked my questions in here.
I will absolutely appreciate whoever gives enlightening piece of favor!
*As a side note, people around me, students, and TA's can't even understand the question.So, it has become really important for me to know the answer!
It's true that the question is not very clear. But look, there is something simple going on here. If we have an equation: $$\text{something} = \text{something else}$$
Then surely we also have the equation: $$\text{derivative}(\text{something}) = \text{derivative}(\text{something else})$$
So if you accept that we can differentiate one side of the equation, then you must accept that we can differentiate both sides and set them equal to one another.
Do you accept this? You say that you understand that we treat $y$ as a function of $x$, but you ask what that function is. This misses the point. Implicit differentiation is useful precisely when we don't know exactly what function we're dealing with. For example, suppose that I'm looking at this equation:
$$e^y \sin y = x^3 + x^x$$
Solving for $y$ would be a nightmare (if it were even possible). But we can still obtain information about the derivative of $y$, just by differentiating both sides using the chain rule (do this; it's a good exercise).
In some cases, we may be able to find the slope of a curve at certain points without ever knowing a parameterization of the curve.
All of this hinges on whether or not we can think of $y$ in the above equation as a function of $x$. So? Can we?
(Note: MrSlunk just posted a link to a nice answer that discusses this final question in detail.)