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Assuming, naively, that one acquires the nth derivative of a function by repeatedly differentiating and finding a pattern. Thus one gets $f^{(n)}(x)=g(x,n)$. I have a few questions about this situation.

The first derivative at a point is the slope of the line tangent to the curve at the point. The second derivative measures the concavity of the graph of a function. Are there corresponding "explanations" for higher order derivatives?

The first integral measures the area between the curve and the x-axis. Are there similar "explanations" for higher integrals?

If one has the function $g(x,n)$, does it make sense to put a non-integer real number or imaginary number on the place of $n$? I know there exists a rigorous field of fractional calculus, but I know next to nothing about it.

If one has $g(x,n)$, when will $g(x,0)=f(x)$? How about when will $g(x,-1)=\int f(x) dx ?$

If one replaces $n$ in $g(x,n)$ with a fraction, one will get a proper function. What is this function, what does it "mean", does it have an "explanation" similar to first or second derivative?

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@FlybyNight I know that, my point, I think, is that it is possible to interpret derivative in this way. Whether or not it makes sense to do so in the bigger picture, is a different question. –  Valtteri Feb 6 '13 at 21:33
@FlybyNight sorry about that, was on a cruise. The wikipedia article was known to me, but thank you anyway. –  Valtteri Feb 12 '13 at 18:00

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