Ok, got it. Thanks a lot for explaining. My partial diffferential calculus is very rusty, and I was confusing between partial derivative chain rule (where it does not make sense to cancel out the dx terms in each partial derivative product, since they are different partial derivative terms being multiplied) with the univariate chain rule, where we can cancel out such terms.
For others who land here with similar confusion, This article 10 about chain rule for partial derivative (see case 2) also helped clarify.
Thanks again for clarifying - this was very baffling - now it is clear.
Anand