Derivative of differential equation

Question

The third degree seems to be incorrect when I use the derivative formula. Could someone guide me $$\begin{align} &y'=-2xy^2\\ &y''=-2[y^2+xyy']\\ &y'''=-2[2yy'+yy'+(xy')^2+xyy'']\\ &y''''=? \end{align}$$

Answer 1

Note that the derivative of a product of $n$ functions is the sum of $n$ products where we differentiate just one factor at the time (see also The derivative of a product of more than two functions?). For example for $3$ functions, $$(fgh)'=f'gh+fg'h+fgh'.$$ Therefore, it should be $$y''=-2[y^2+\color{blue}{2}xyy']$$ $$y'''=-2[2yy'+\color{blue}{2}yy'+\color{blue}{2x(y')^2}+\color{blue}{2}xyy''].$$ What is the fourth-derivative $y''''$?