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EDIT: I think an answer to your questions (and a lot about answers) are first question is explained in the papers:

  • P.-A. Meyer, Qu'est ce qu'une différentielle d'ordre $n$, Exposition. Math. 7 (1989), 249–264.
  • Laksov, Dan; Thorup, Anders, These are the differentials of order $n$. Trans. Amer. Math. Soc. 351 (1999), no. 4, 1293–1353. Freely available online.

Quote from the second:

The higher order differentials were part of the folklore of mathematics up to the end of the previous century, and formulas like $d^2f=f'_xd^2x+f'_yd^2y+{f''_{x^2}}dx^2+2{f''_{xy}}dxdy+{f''_{y^2}}dy^2$ can be found in most classical calculus books. (...) the higher order differentials vanished (...) because the extensive user of exterior differentials led mathematicians to believe that $d^2$ should always be zero.

(...)

Let $C_n:=\mathcal{C}^\infty(T^nX)$. The differential of $\mathcal{C}^\infty$- functions on $T^nX$ can be viewed as a k-linear map $d:C_n\rightarrow C_{n+1}$, and we obtain a sequence of linear maps...

Then $\Omega^n$ is defined as a suitable submodule of $C_n$, and there is a product $\Omega^p \otimes \Omega^n \rightarrow \Omega^{p+n}$.Omega^{p+n}$ such that $d(\omega\cdot\pi)=d\omega\cdot\pi + \omega\cdot d\pi$.
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Your

I think your questions (and a lot about answers) are explained in the papers:

  • P.-A. Meyer, Qu'est ce qu'une différentielle d'ordre $n$, Exposition. Math. 7 (1989), 249–264.
  • Laksov, Dan; Thorup, Anders, These are the differentials of order $n$. Trans. Amer. Math. Soc. 351 (1999), no. 4, 1293–1353. Freely available online.

Quote from the second:

The higher order differentials were part of the folklore of mathematics up to the end of the previous century, and formulas like $d^2f=f'_xd^2x+f'_yd^2y+{f_{x^2}}''dx^2+2{f_{xy}}''dxdy+{f_{y^2}}''dy^2$d^2f=f'_xd^2x+f'_yd^2y+{f''_{x^2}}dx^2+2{f''_{xy}}dxdy+{f''_{y^2}}dy^2$ can be found in most classical calculus books. (...) the higher order differentials vanished (...) because the extensive user of exterior differentials led mathematicians to believe that $d^2$ should always be zero.

(...)

Let $C_n:=\mathcal{C}^\infty(T^nX)$. The differential of $\mathcal{C}^\infty$- functions on $T^nX$ can be viewed as a k-linear map $d:C_n\rightarrow C_{n+1}$, and we obtain a sequence of linear maps...

Then $\Omega^n$ is defined as a suitable submodule of $C_n$, and there is a product $\Omega^p \otimes \Omega^n \rightarrow \Omega^{p+n}$.
show/hide this revision's text 2 fixed the formula

Your questions (and a lot about answers) are explained in the papers:

  • P.-A. Meyer, Qu'est ce qu'une différentielle d'ordre $n$, Exposition. Math. 7 (1989), 249–264.
  • Laksov, Dan; Thorup, Anders, These are the differentials of order $n$. Trans. Amer. Math. Soc. 351 (1999), no. 4, 1293–1353.

Quote from the second:

The higher order differentials were part of the folklore of mathematics up to the end of the previous century, and formulas like $d^2f=f'_xd^2x+f'yd^2y+{f{x^2}}''dx^2+2{f_{xy}}''dxdy+{f_{y^2}}''dy^2$ d^2f=f'_xd^2x+f'_yd^2y+{f_{x^2}}''dx^2+2{f_{xy}}''dxdy+{f_{y^2}}''dy^2$ can be found in most classical calculus books. (...) the higher order differentials vanished (...) because the extensive user of exterior differentials led mathematicians to believe that $d^2$ should always be zero.

Let $C_n:=\mathcal{C}^\infty(T^nX)$. The differential of $\mathcal{C}^\infty$- functions on $T^nX$ can be viewed as a k-linear map $d:C_n\rightarrow C_{n+1}$, and we obtain a sequence of linear maps...

Then $\Omega^n$ is defined as a suitable submodule of $C_n$, and there is a product $\Omega^p \otimes \Omega^n \rightarrow \Omega^{p+n}$.
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