In a lecture(see after 0:09:00 in the video) by Dr. Tadashi Tokieda on topology and geometry, it is being argued by him that, if we have two manifolds $M$ and $N$ with dimensions $m,n$ and boundaries $\partial M, \partial N$ respectively, then the boundary $\partial (M\times N)$ of the product manifold $M\times N$, is given by $$\partial (M\times N)=(\partial M \times N)\cup (M\times \partial N)$$ He continues to argue that the boundary operator $\partial$ acts like Leibniz law for differentiation and claims that the analogy has something to do with the so called "de Rham cohomology"
My question: What is the connection between Leibniz law , boundary operator and the de Rham cohomology?