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I'm just curious if it is possible to integrate or derive a function so that it becomes a equation that is not a function like a hyperbola, circle, or something else?

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What do you mean by becomes a non-function? – Sigur Feb 4 '13 at 23:52
@Throsby: As currently written, your question is not making sense. Is it possible to clarify what you mean? Regards – Amzoti Feb 5 '13 at 0:16
Sobolev spaces include functions whose (weak) derivatives are not functions according to the classical definition. Is this what you mean? – AndreasT Feb 5 '13 at 0:21
A circle is not a function. A circle is not even the graph of a function. – Gerry Myerson Feb 5 '13 at 2:32

I don't know whether this is what OP has in mind, but here goes:

If a function is given by a formula $y=f(x)$, using only the familiar functions of intro calculus such as powers, exponentials, logarithms, trig and inverse trig, and in finite terms (no infinite sums or products or compositions), then its derivative is given by the same kind of formula, except that there may be a few points where the derivative is not defined.

This is not true for antiderivatives, as, e.g., $\int x^{-1}e^x\,dx$ cannot be given in terms of the familiar functions listed above.

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