I've got a function $g$ which is the result of nesting a sequence of functions $f_1 \cdots f_n$, $$ g_n(x,y) = f_1(f_2(f_3(\cdots f_{n-1}(f_n(x,y),y),y),y),y) $$
I'm looking to find the second and third partial derivatives of $g_n(x,y)$ w.r.t y $$ \frac{\partial^2 g_n}{\partial y^2}, \frac{\partial^3 g_n}{\partial y^3} $$
I've determined that the first derivative is $$ \frac{\partial g_n}{\partial y} = \frac{\partial f_1}{\partial y} + \frac{\partial f_1}{\partial f_2} \left( \frac{\partial f_2}{\partial y} + \frac{\partial f_2}{\partial f_3} \left(\cdots \left( \frac{\partial f_{n-1}}{\partial f_n} \left( \frac{\partial f_n}{\partial y} \right) \right) \right) \right) \\ \frac{\partial g_n(x,y)}{\partial y} = \sum_{i=1}^n \frac{\partial F_i}{\partial y} \prod_{j=1}^{i-1} \frac{\partial F_{j}}{\partial F_{j+1}} $$
What about the second or third partial derivative w.r.t y?