Chain rule

Definition

1 dimensional case

Here $f(x)$ and $g(t)$ are functions:

$$\frac{d}{dt}\Big[f\circ g\Big]=\left.\frac{df}{dx}\right|_{g(t)}\frac{dg}{dt}$$

Then:

$$\frac{d^2}{dt^2}\Big[f\circ g\Big]=\frac{d}{dt}\Big[\left.\frac{df}{dx}\right|_{g(t)}\frac{dg}{dt}\Big]$$ $$=\left.\frac{df}{dx}\right|_{g(t)}\cdot\frac{d}{dt}\Big[\frac{dg}{dt}\Big]+\frac{dg}{dt}\cdot\frac{d}{dt}\Big[\left.\frac{df}{dx}\right|_{g(t)}\Big]$$ $$=\left.\frac{df}{dx}\right|_{g(t)}\cdot\frac{d^2g}{dt^2}+\frac{dg}{dt}\cdot\frac{d}{dt}\Big[\left.\frac{df}{dx}\right|_{g(t)}\Big]$$

That is:

$$\frac{d^2}{dt^2}\Big[f\circ g\Big]=\left.\frac{df}{dx}\right|_{g(t)}\cdot\frac{d^2g}{dt^2}+\frac{dg}{dt}\cdot\frac{d}{dt}\Big[\left.\frac{df}{dx}\right|_{g(t)}\Big]$$

Little can be done about

$$\frac{d}{dt}\Big[\left.\frac{df}{dx}\right|_{g(t)}\Big]$$ at this point. It is "the change in the rate of change of f with respect to x taken at g(t) with respect to t" which has little to do with $$\frac{d^2f}{dx^2}$$ computationally.