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What you're seeing is the displayed notation for symbolic partial derivatives. D[0] means "apply the derivative operator with respect to variable number 0" and D[0,0] represents two successive derivatives with respect to the variable. Observe:
sage: f = function('f', x)
sage: f.diff(x)
D[0](f)(x)
sage: f.diff(x).diff(x)
D[0, 0](f)(x)
