How to calculate the derivative/gradient of this scalar product?

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in my exam prep I have the following:

enter image description here

I need to calculate the gradient of this function f. So I start using the chain rule but how do I get the derivative of <x,wi>?

Why does the x of the derivative disappear? I tried to use: <a,b> = a'b+b'a.

But wouldn't this give me x+w? Is this formular wrong to use in this context?

The solution only states this but not how to get there:

enter image description here

Thanks in advance!