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Yet Another Proof of Sylvester's Determinant Identity

In 1857 Sylvester stated a result on determinants without proof that was recognized as important over the subsequent century. Thus it was a surprise to Akritas, Akritas and Malaschonok when they found only one English proof - given by Bareiss 111 years later! To rectify the gap in the literature these authors collected and translated six additional proofs: four from German and two from Russian. These proofs range from long and "readily understood by high school students" to elegant but high level. We add our own proof to this collection which exploits the product rule and the fact that taking a derivative of a determinant with respect to one of its elements yields its cofactor. A differential operator can then be used to replace one row with another.