If U=F(x,y,z), and z=f(x,y), then find the formula of $\frac{∂^2U}{∂x^2}$ in terms of derivatives of "F" and derivative of "z" repectively. I have been attempting this question in many ways but I haven't yet reached any conclusion
As we are already given to find $\frac{∂^2U}{∂x^2}$
so I know that $\frac{∂U}{∂x}$=$\frac{∂F}{∂x}$
but how should I represent it as a derivative of z ?
 A: One thing that is to be noticed is that $z$, which is one of the "independent" variables on which the value of $U$ depends, it is itself a function of variables $x$ and $y$. Thus, if we are to write $\delta U/\delta x$ in terms of derivatives of $F$, we get
$$\frac{\delta U}{\delta x}=\frac{\delta F}{\delta x}+\frac{\delta F}{\delta z}\cdot\frac{\delta z}{\delta x},$$
which further gives,
$$\frac{\delta^2 U}{\delta x^2}=\frac{\delta^2 F}{\delta x^2}+\frac{\delta^2 F}{\delta x\ \delta z}\cdot\frac{\delta z}{\delta x}+\frac{\delta F}{\delta z}\cdot\frac{\delta^2 z}{\delta x^2}.$$
Thus, in standard notation, this can also be written as
$$U_{xx}=F_{xx}+F_{zx}\cdot z_x+F_z\cdot z_{xx}.$$
A simple way to go about finding partial derivatives of a dependent variable is to prepare a little chart to show variable relationships. So, in this case, we have that $U$ is a function of $x$, $y$ and $z$. So, we write $U$ at the extreme right of the diagram, and then write $x$, $y$ and $z$ on a (roughly) vertical line to the left of where we wrote $U$ earlier. We draw arrow-headed lines connecting each of $x$, $y$ and $z$ to $U$, showing that $U$ depends on these variables. Now, we have been given in the question that $z$ itself further depends upon $x$ and $y$. To account for that, we write $x$ and $y$ again to the left of $z$, and draw lines connecting $x$ and $y$ to $z$. This shows that $z$ depends upon $x$ and $y$. Now, the task of finding partial derivative gets reduced to finding all the "paths" in this diagram from $U$ back to $x$, and using the chain rule along each. We get two paths connecting $x$ to $U$: one directly, and one through $z$. The direct path gives us $\dfrac{\delta F}{\delta x}$ and the other one through $z$ gives us $\dfrac{\delta F}{\delta z}\cdot\dfrac{\delta z}{\delta x}$. We then simply add them up.

