I can't work out what WolframAlpha is doing wrong with this query:"derivative of (vector{1,2} dot vector{x,x^2}) with respect to x"

It can do the dot product properly on its own:

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it looks like it's just ignoring large parts of my query. "derivative of {x,x^2}" produces the same result:
I still can't see how you get from {x,x^2} to x^3+x^2.

@christianp I forgot to add in my previous response that specifying the differentiation variable is often a good idea instead of making Alpha guess, if you prefer natural language input:

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