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What about

sage: var('t u v')
(t, u, v)
sage: x = function('x')(t)
sage: y = function('y')(t)
sage: w(u,v) = u^2 + v^2
sage: diff(w(x,y), t)
2*x(t)*diff(x(t), t) + 2*y(t)*diff(y(t), t)