The differential $d(d(f(x,y)))$

Question

I have no idea if the question is going to be duplicated or not; what is the differential of the second order? If $\varphi(x,y)$ is a nice function that can be integrated and differentiated a desired number of times, then what is $d^2 \varphi$? Basically, it is explained to be the differential of the differential, so we apply recursion to the equality $$d\varphi = \frac{\partial \varphi}{\partial x}dx+\frac{\partial \varphi}{\partial y}dy$$ That is, $$d^2 \varphi =\frac{\partial}{\partial x}\left(\frac{\partial \varphi}{\partial x}dx+\frac{\partial \varphi}{\partial y}dy\right)dx+\frac{\partial}{\partial y}\left(\frac{\partial \varphi}{\partial x}dx+\frac{\partial \varphi}{\partial y}dy\right)dy$$ As you can see, I used the first formula for the expression on the right side of it. Following the same idea, we can acquire the expression for $d^3\varphi$. Wikipedia says that the exact differential can be defined as a map?? But my question is not related to it but related to why in books, the differentials are of the general form: $$d^n \varphi =\left(\frac{\partial \varphi}{\partial x}dx+\frac{\partial \varphi}{\partial y}dy\right)^n$$ I would like to apologize for such an unsuitable question but I do wish to discover the tiny truth about these expressions. We deal with TOTAL DIFFERENTIAL, not with exact - I mixed concepts...