I just graduated and now feel masochistically inclined to torture myself with diff eq. My question relates to differentiating terms in implicit functions using the product rule. I understand how to use the product rule with two differentiable functions of x. I do not, however, quite understand how to use this rule when the terms involve y and x.
I have attached a screenshot of the problem. I have highlighted in red the term whose origin I do not quite understand.
f(x) = y^4 + 2x²y² + 6x² = 7
u = 2x²y²
v = y²
The derivative of 2x²y² w.r.t x is 2y(dy/dx). But why?
Question: Why does dy/dx appear from u(dv/dx)? Why is the derivative not simply 2y?
Thank you for any help,