Taylor Expansion (multivariate)

Hi, I need some help understanding the taylor expansion of a function . I'm particularly confused with the 3rd term on the RHS. Why is there a partial derivative there? When I expand it, I get the same as whats in the image but no multiplication with partial derivatives.
Thanks


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The notation you're using here is terrible. Where did you get this expression? Use for the function of 3 variables. The third term of the series comes from the Hessian matrix. Do you what that is?

Thanks for replying ojones. Turns out the partial derivative multiplying the 3rd term and other terms were a typo. That's odd since this typo was present in multiple papers.