The $n$th derivative of $f(x) = \frac 1 x$ is $$f^{(n)}(x) = (-1)^n n! \frac 1 {x^{n+1}}$$ so that e.g. $$f^{(10)}(x) = \frac {10!} {x^{11}}.$$
While this formula is easy to calculate, it's somewhat mystifying. Why do the higher derivatives grow so rapidly? Compare this with $g(x) = x^m$, where the higher derivatives quickly go to $0$.
Graphically, there's nothing I can see in the graph of $f$ to suggest these extremely large higher order derivatives. After a certain point, the graph just seems flat (to within the limits of the eye's ability to recognize).
Intuitively, I'd associate such large derivative to some type of radical change or oscillation; but there's nothing I can see of the sort, only a pretty flat curve.
So: Why do the higher order derivatives of $\frac 1 x$ grow extremely large?