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.
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.
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.