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$\mathrm{with}\left(\mathrm{DifferentialAlgebra}\right)\:$$\mathrm{with}\left(\mathrm{Tools}\right)\:$

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$R\u2254\mathrm{DifferentialRing}\left(\mathrm{derivations}=\left[x\,y\right]\,\mathrm{blocks}=\left[u\,v\right]\right)$

${R}{\u2254}{\mathrm{differential\_ring}}$
 (1) 
The triangular case: the least common derivative of the two leading derivatives is different from both of them.
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$\mathrm{DeltaPolynomial}\left(u\left[x\right]v\,u\left[y\right]\,R\right)$

The nontriangular case: the leading derivative of the second argument is a derivative of the leading derivative of the first one.
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$\mathrm{DeltaPolynomial}\left({u\left[x\right]}^{2}4u\,u\left[x,x\right]\,R\right)$
