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the derivative of f(x) with respect to y means, how much does f(x) change as y changes. But f(x) is not a function of y (it is a function of x) so it doesn't change as y changes. The change is 0, hence the derivative is 0.
But y==f(x) so if I want dy/dy, that would be the change in y as y changes. If y changes by some amount, y will also change by some amount. So this really amounts to df(x)/df(x) which would just be 1. Why does this logic fail?
Actually, I was wrong. Well initially I was wrong. The answer should be one for the reason I stated above. Also, I checked on wolfram alpha.