Given f(x,y), x(t,u) and y(t,u), how do I intuitively calculate the second partial derivative with respect to u? I'm a bit confused when dealing with higher order derivatives. Suppose we have a function f(x,y) where x(t,u) and y(t,u).
Can someone help me understand how I should think about solving $\frac{\delta^2f}{\delta x^2}$ or $\frac{\delta^2f}{\delta u^2}$?
I've read my litterature but at some point I started getting confused by the indexing, and I feel now that I don't understand it fully anymore.
I'll walk you through my though process.
When doing the second partial derivative of $\frac{\delta^2f}{\delta u^2}$, my first though is to do $\frac{\delta f}{\delta u}(\frac{\delta f}{\delta u})$, where $\frac{\delta f}{\delta u}=\frac{\delta f}{\delta x}\frac{\delta x}{\delta u}+\frac{\delta f}{\delta y}\frac{\delta y}{\delta u}$ which can be written as: $\frac{\delta x}{\delta u}*f_1+\frac{\delta y}{\delta u}*f_2$, correct? And now to get $\frac{\delta^2f}{\delta u^2}$ one would simply need to compute the result of $\frac{\delta f}{\delta u}(\frac{\delta x}{\delta u}*f_1+\frac{\delta y}{\delta u}*f_2)$, correct?
It is here I start to get confused. When reviewing an example from the litterature they seem to do the opposite of an exercise that I later solved, and my head starts to spin because of all the indexing and so on... But this is what I think I should do...
$\frac{\delta x}{\delta u}(\frac{\delta x}{\delta u}*f_{11}+\frac{\delta x}{\delta t}*f_{12})+\frac{\delta y}{\delta u}(\frac{\delta y}{\delta u}*f_{21}+\frac{\delta y}{\delta t}*f_{22})$
And why is becuase when I review the indexing, $f_{11}$ tells me that I should derivate with respect to the function $f$s first variable, $x$, and again with respect to the first variable of $x$ which is $u$, but is my thinking correct?
So my question is, what is $\frac{\delta^2f}{\delta x^2}$ and $\frac{\delta^2f}{\delta u^2}$ and how can I memorize the steps requiered to compute these things in the future?
EDIT:
Can I perhaps use the fact that $\frac{\delta^2 f}{\delta u^2}=(\frac{\delta f}{\delta x}\frac{\delta x}{\delta u}+\frac{\delta f}{\delta y}\frac{\delta y}{\delta u})(\frac{\delta f}{\delta x}\frac{\delta x}{\delta u}+\frac{\delta f}{\delta y}\frac{\delta y}{\delta u})$  and then simply expand with taking the products of the terms?
If I expand upon this I get:
$\frac{\delta f}{\delta x}\frac{\delta x}{\delta u}(\frac{\delta f}{\delta x}\frac{\delta x}{\delta u}+\frac{\delta f}{\delta y}\frac{\delta y}{\delta u})+\frac{\delta f}{\delta y}\frac{\delta y}{\delta u}(\frac{\delta f}{\delta x}\frac{\delta x}{\delta u}+\frac{\delta f}{\delta y}\frac{\delta y}{\delta u})$.
I can evaluate $\frac{\delta f}{\delta x}\frac{\delta f}{\delta x}=f_{11}$ and the others also get evaluated to such terms, which simplifies it to:
$\frac{\delta x}{\delta u}(f_{11}\frac{\delta x}{\delta u}+f_{12}\frac{\delta y}{\delta u})+\frac{\delta y}{\delta u}(f_{21}\frac{\delta x}{\delta u}+f_{22}\frac{\delta y}{\delta u})$
This is however in contradiction with what I guessed before. Furthermore, if this is correct (which I percieve it to be), how do I then now proceed to derivate these terms now? Given that I have formulas for $x(t,u)$ and $y(t,u).$
 A: $$\frac{\partial f}{\partial u} = \frac{\partial f}{\partial x}\frac{\partial x}{\partial u} + \frac{\partial f}{\partial y}\frac{\partial y}{\partial u}$$
$$\implies \frac{\partial^2 f}{\partial u^2} = \frac{\partial}{\partial u}\left(\frac{\partial f}{\partial x}\right)\frac{\partial x}{\partial u} + \frac{\partial f}{\partial u}\frac{\partial ^2 x}{\partial u^2} + \frac{\partial}{\partial u}\left(\frac{\partial f}{\partial y}\right)\frac{\partial y}{\partial u} + \frac{\partial f}{\partial u}\frac{\partial ^2 y}{\partial u^2}$$
$$\implies \frac{\partial^2 f}{\partial u^2} =  \left(\frac{\partial ^2f}{\partial x^2}\frac{\partial x}{\partial u}+ \frac{\partial ^2f}{\partial x \partial y}\frac{\partial y}{\partial u}\right)\frac{\partial x}{\partial u} + \frac{\partial f}{\partial u}\frac{\partial ^2 x}{\partial u^2} + \left(\frac{\partial ^2f}{\partial x \partial y}\frac{\partial x}{\partial u}+ \frac{\partial ^2f}{\partial y^2}\frac{\partial y}{\partial u}\right)\frac{\partial y}{\partial u} + \frac{\partial f}{\partial u}\frac{\partial ^2 y}{\partial u^2}$$
A: Let $F(t,u)=f(x(t,u),y(t,u))$.
Then
$F_u=f_x x_u  + f_y y_u$ by the Chain Rule.
Note that this is a sum of two products, so we use the Sum, Product rules to get
$$
F_{uu}=
(f_x)_u x_u +f_x x_{uu} 
+ (f_y)_u y_u + f_y y_{uu}.
$$
Now use the Chain Rule twice more to get
$$
F_{uu}=
(f_{xx} x_u +f_{xy} y_u ) x_u +f_x x_{uu} 
+ (f_{yx} x_u +f_{yy}y_u   ) y_u + f_y y_{uu}.
$$
Collect the terms and we  have
$$
F_{uu}= f_{xx}x_u^2 +2 f_{xy} x_u y_u + f_{yy} y_u^2 +f_x x_{uu} + f_y y_{uu}.
$$
