$d$-dimensional Gaussian random vector pdf

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I am reading this paper: https://www.fil.ion.ucl.ac.uk/~wpenny/course/bayes.pdf On page $4$ ( point $10.4$ ), there is a statement about the pdf of $x,$ which I cannot understand. How is this equation derived? I tried to calculate it myself multiple times and got a different result. Can anyone prove it or give a source for a proof? Thanks :)