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I want to replace the variable x in the result, not x in unrelated x[i].

ω[n_] := Product[x - x[i - 1], {i, 1, n}]

D[ω[3], x] /. x -> x[k]

In the code above, I want to get a result like (x[k] - x[0])*(x[k] - x[1]) + (x[k] - x[2])*(x[k] - x[1]) + (x[k] - x[0])*(x[k] - x[2]). how can I solve this problem skillfully?

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You can use Replace with the option Heads -> False:

Replace[D[ω[3], x], x -> x[k], All, Heads -> False]
 (-x[0] + x[k]) (-x[1] + x[k]) + (-x[0] + x[k]) (-x[2] + x[k]) + 
  (-x[1] + x[k]) (-x[2] + x[k])

Alternatively, you can use ReplaceAll with the replacement rule {p_x :> p, x -> x[k]}:

D[ω[3], x] /. { p_x :> p, x -> x[k]}
 (-x[0] + x[k]) (-x[1] + x[k]) + (-x[0] + x[k]) (-x[2] + x[k]) + 
  (-x[1] + x[k]) (-x[2] + x[k])
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