Learning multiple tasks using Manifold regularization: Question about the paper.

I was reading the paper "Learning multiple tasks using Manifold regularization"(http://www.umiacs.umd.edu/~arvinda/mysite/papers/nipsMTL.pdf). This paper appeared in NIPS 2010. I was wondering if someone in the community has already read this. I tried to read the background reference paper, "Dimensionality reduction using Kernel Map Methods" (http://www.cs.utah.edu/~sgerber/research/kmm_iccv09.pdf) to gain some intuition.

Specifically I am trying to justify (to myself) how the gradient of the projection distance reduces to the more simplified form from Equation 8 to Equation 9 in Section 3.1. Any suggestions in the geometric interpretations would be helpful.