In my textbook there are two questions on finding second order derivatives for a function $f(x(u, v), y(u, v)$.
In the first question I have to take:
$ (\frac{\partial}{\partial x})(y\frac{\partial f}{\partial x}) $
$ = \frac{\partial f}{\partial x} \frac{\partial y}{\partial y} + y \frac{\partial^2f}{\partial x^2} $ (applying the chain rule)
$ = \frac{\partial f}{\partial x} + y \frac{\partial^2f}{\partial x^2} $
Then in the second question I have to take:
$ (\frac{\partial}{\partial x})(\frac{\partial f}{\partial x}u) $
$ = u\frac{\partial^2f}{\partial x^2} $ (not applying the chain rule)
I do not understand why the chain rule must be applied in the first case but not the second, and there is no explanation in my textbook. I think it's to do with the first case being differentiating wrt the a variable from the same coordinate system, and the second being from different coordinate systems, but I can't get any further that that. Any help would be greatly appreciated!
Thank you!