Please help soving 3rd part of this question - I have tried taylor's expansion but I am not sure if this is the right answer

Dr.Peterson said:
The notation here seems very odd. I think they are talking about the directional derivative, but in the questions the subscripts seem to be just sets of variables rather than vectors.
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Odd indeed. Is it possible that in the case of [imath]\nabla_{x,y} f[/imath] they mean [imath]\mathbf = (x,y,0,0,0)[/imath]?
Dr.Peterson said:
Probably you are expected to use your second answer to help with the third.
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Is the numbering zero-based?
 
blamocur said:
Odd indeed. Is it possible that in the case of [imath]\nabla_{x,y} f[/imath] they mean [imath]\mathbf = (x,y,0,0,0)[/imath]?
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My impression is that the notation means "the gradient of f, considered as a function of only x and y, with the other variables treated as parameters"

blamocur said:
Is the numbering zero-based?
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I'm not sure what you're saying here, but I was talking about the answer in the second blue box ([imath]\nabla_{a,b,c} f(x,y;a,b,c)[/imath]) being used to find the answer in the third blue box (the approximation).
 
Dr.Peterson said:
I'm not sure what you're saying here, but I was talking about the answer in the second blue box
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My mistake, and my apologies. I saw four bullet items and somehow missed the fact that only three of them are questions.
 
Dr.Peterson said:
My impression is that the notation means "the gradient of f, considered as a function of only x and y, with the other variables treated as parameters"
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It looks to me that those are equivalent definition since by [imath](x,y,0,0,0)[/imath] I meant a (direction) vector, not a point.
 
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