I'm reading through Bamberg and Sternberg, and I'm on Chapter 5. It has the attached passage. I understand it up until it says we can substitute our $dy=15x^2dx$ into the $d(y^2)=2y\space dy$ equation.
It states that $dy$ replaces $h$ as our displacement, since before it was made clear that we could write $d\alpha_y(h)=\alpha'(y)h,$ so this is merely a change in notation. However, all we did in the bottom was define a function $y(x)=5x^3+1$ whose differential $dy$ so happeend to have the same name as what we replaced $h$ with. So, how come we can just substitute this in to result in an example of the chain rule?
To be clear, I mean: if the $dy$ in $d\alpha=\alpha'dy$ is just a real number and if the $dy$ in $dy=15x^2dx$ is instead a function of $x,$ how can we substitute the second instance of $dy$ into the first instance of $dy$, just because we named them the same?
