the meaning of the notation behind derivatives In Calculus, we are taught $\frac{dy}{dx}$ (first derivative), $\frac{d^2y}{dx^2}$ (second derivative), ..., $\frac{d^ny}{dx^n}$ ($n$-th derivative) respectively. Am wondering does the notion of a $\frac{d^my}{dx^n}$ ($m\neq n$) exist? what does that mean? any approachable, understandable reference you can cite for me (a non-mathematician, math hobbyist at best) to learn these concepts clearly from basic principles? Thank you. 
 A: The notation is reminiscent of that for differences. The thing is that the first difference of a function $\Delta f=f(x+\Delta x)-f(x)$ is typically of the same order of $\Delta x$ (it is so at points where $f$ is differentiable and $f'(x)\neq 0$. If you consider the second difference $\Delta^2f=\Delta f(x+\Delta x)-\Delta f(x)$ (a difference of differences) it is a smaller infinitesimal, typically of the order of $\Delta x^2$, etc. This is why the interesting limits are the limits of $\Delta f/\Delta x$, $\Delta^2f/\Delta x^2$, etc. which are precisely the corresponding derivatives. If you divide $\Delta^2f/\Delta x^3$ you get infinity in the limit, if you divide $\Delta^2f/\Delta x$ you get zero because the infinitesimals have different orders. This is why $d^nf/dx^m$ with $n\neq m$ is not an interesting object, unless.. at a critical point you have $f'(x)=0$ and $\Delta f$ happens to be typically quadratic. Then, it makes sense to look at $\lim \Delta f/\Delta x^2$ which is equal to $f''(x)/2$ (just do Taylor expansion).  At certain singular points such mixed objects are finite and make perfect sense. 
Nowadays they teach you that $df/dx$ is just a symbol and not a fraction. This is really sad, since the connection to differences is lost, not to mention the intuition behind the concept. There are fashions in Math like in anything. $df/dx$ is not a fraction now, but it was a fraction for the Mathematicians who created all the fundamentals of infinitesimal calculus in the XVIII and XIX centuries.
