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In the latter case, $y$ is buried, in some sense; if you can "dig it out", then you can write it explicitly.
A bunch of you started looking at derivatives. It's not about derivatives. It's just about how a function is expressed.
Example: If I describe to you my philosophy of life, I'm being explicit (then you watch to see if I follow it!); alternatively, you can see how I behave and deduce my philosophy of life, which is given implicitly.
The use of these terms mathematically coincides with their use in common speech.
Some things can be written implicitly more easily than explicitly, e.g. the circle of radius $r$ centered at the origin:
The graph of a circle fails the VLT -- it must be represented as two (or more) explicit functions, but can be represented very nicely via the implicit representation above.
Another way to give $y$ implicitly is to say that $y$ is one coordinate of a circle of radius $r$! We just give $y$ in words.