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?