Math Notation On Chain Rule of Second Derivatives

Question

I have a math notation question. Say I have a function $y=F(z)$, where $z=x-vt$. Now I want to find $\dfrac{\partial^2y}{\partial{x^2}}$ and $\dfrac{\partial^2y}{\partial{t^2}}$. First, we find that$$\frac{\partial y}{\partial x} = \frac{\partial F}{\partial z} \frac{\partial z}{\partial x} = \frac{\partial F}{\partial z}.$$Is it correct to show that$$\frac{\partial^2y}{\partial x^2} = \frac{\partial}{\partial x}\left(\frac{\partial F}{\partial z}\right) = \frac{\partial^2 F}{\partial z^2}=\frac{\partial F}{\partial z}\frac{\partial z}{\partial x}\frac{\partial F}{\partial z}\frac{\partial z}{\partial x},$$or is that incorrect notation? I hope that isn't vague! I am specifically not certain about the correctness of the last part. Thank you.

Answer 1

I think what you wrote down gets the point across, but I prefer something more terse:

Let $F$ be twice differentiable and $z$ be once differentiable with $z^{\prime} = 1$. Let $G=F\circ z$. By chain rule, $G^{\prime}=z^{\prime}\cdot F^{\prime}\circ z=F^{\prime}\circ z$. Similarly, $G^{\prime\prime}=z^{\prime}F^{\prime\prime}\circ z=F^{\prime\prime}\circ z$.