Is there any other evidence of this mathematical concept existing in Babylon before Pythagoras?
As Wikipedia observes, the Plimpton 322 tablet
… lists two of the three numbers in what are now called Pythagorean triples, i.e., integers a, b, and c satisfying a2 + b2 = c2
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In addition to the Plimpton 322 tablet we have:
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This has a diagram of a square with diagonals. One side of the square is labelled '30' (in Babylonian numerals, base 60). Across the centre on the diagonal we see the numbers '1, 24, 51, 10' and '42, 25, 35' (also in Babylonian numerals).
Not only does this show an understanding of what we call 'Pythagoras's theorem', it also shows that the ancient Babylonians knew a pretty good approximation to the value of √2.
(For more detail, see the page Pythagoras's theorem in Babylonian mathematics from the School of Mathematics and Statistics, University of St Andrews, cited below)
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Excavated in 1936, and published by E.F Bruin in 1950 (Quelques textes mathématiques de la mission de Suse), this shows a problem calculating the radius of a circle through the vertices of an isosceles triangle.
Discovered in an excavation close to Baghdad in 1962, and dating to the reign of Ibalpiel II of Eshunna (c 1750 BCE), this deals with calculating the dimensions of a rectangle where the area & diagonal are known.
However, none of this was presented in the form of a 'theorem'. We have evidence for the Babylonians using the concepts in particular cases, but no evidence for a general rule or formal proof.
There is a discussion of the evidence from the tablets described above on the page Pythagoras's theorem in Babylonian mathematics from the History of Mathematics archive of the School of Mathematics and Statistics, University of St Andrews.
[Incidentally, the ancient Egyptians also knew the concepts underlying 'Pythagoras' Theorem' long before the time of Pythagoras. The solution to one of the problems preserved in the Berlin Papyrus 6619 is a Pythagorean triple (although whether that, in itself, implies an understanding of the concepts remains a matter of debate). More convincingly, we have evidence that they used knotted ropes in the ratio 3-4-5 to form right-angled triangles for laying out stonework.]
Any evidence of Pythagoras deriving his work from Babylonian mathematics?
That is not to say that Pythagoras didn't know about Babylonian mathematics. He may have done (perhaps via Egypt), but we have no evidence of this.