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So we were learning implicit differentiation a couple of months ago, and I noticed that while for some equations, like ${{x}\over{y}}=1$ can easily be rewritten as $y=x$ and therefore have a very easy derivative to take, some equations, especially those that broke the vertical line rule, were very hard to convert to explicit.

For example, the equation of a circle, $x^2 +y^2=r^2$ can be rewritten as $y=\pm \sqrt{r^2-x^2}$, so my question is: can all implicit formulas be rewritten as one or more explicit equations? If yes, how do we know, but if not, could you provide a counterexample?

Please try to simplify your answers so a high-schooler could get it, but I'll try my best to understand each answer.

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I wish it was true. A counter example is $$y^5 -5y+x = 0$$ Polynomials of degree 5 and above do not have a general formula therefore no explicit forms for $y(x)$ when they are involved in a high order polynomial. This polynomial can be proved not to have an explicit solution.

Another one I have been working on for a long time and to which I can't find anything is : $$x = \tanh^{-1}(y) - 3y$$

Or even $$e^y + y = x$$

Your question could be extended to sequences: can every implicit sequence be rewritten in an explicit form? I wish they did!

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    $\begingroup$ But couldn't $y^5-5y+x=0$ be written as a piecewise explicit function $\endgroup$
    – scrblnrd3
    Dec 23, 2013 at 20:39
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    $\begingroup$ What do you exactly mean by piecewise? $\endgroup$
    – user88595
    Dec 23, 2013 at 20:46
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    $\begingroup$ oh wait no, never mind that wouldn't work $\endgroup$
    – scrblnrd3
    Dec 23, 2013 at 20:52

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