I'm currently in an Advanced Calculus class where we are currently talking about derivatives of functions of several variables, and I came across the following question:
Find the tenth differential, $d^{10}f$, of $f(x,y)=\ln(x+y)$.
After finding a few of the early differentials, I noticed that they followed a similar pattern of the Binomial Theorem for polynomials, e.g., $$d^4f=\frac{-(3!)}{(x+y)^4}(dx^4+4\,dx^3dy+6\,dx^2dy^2+4dx\,dy^3+dy^4),\tag{$\star$}$$ where everything in the parentheses is the fourth row of Pascal's Triangle.
So my question is:
$\textbf{Is there a formal shorthand for writing such differentials?}$
Am I okay to write my fourth differential as I did in $(\star)$?
Could I write an arbitrary $n$th differential as $\frac{(-1)^{n+1}(n-1)!}{(x+y)^n} \sum\limits_{k=0}^n\!{n \choose k}dx^{n-k}\,dy^{k}$ ?
(E.g., $(\star)$ could be written $\frac{(-1)^{5}(3)!}{(x+y)^4} \sum\limits_{k=0}^4\!{4 \choose k}dx^{4-k}\,dy^{k}$ )
Or are both just informal nonsense?
Note: [I am assuming that $dx\,dy=dy\,dx$ since $x+y=0$ is excluded in the domain of $f$, which means all partial derivatives will be continuous on the same domain.]
Writing the $n$-th derivative of a function $f(x)$ as $f^{(n)}(x)$ or $\frac{\mathrm{d}^nf}{\mathrm{d}x^n}$ is commonplace, and your notation is not far from that; I doubt anybody with a mathematical background will have problems understanding what you mean!
Edit: If you are writing a paper or a piece of coursework of some sort, you could define the notation at the beginning just to be safe.