Taking the partial time derivative of a functional

[Arman777]

Let us suppose we have a functional of f such that $f=f((\vec{r}(t),t)$ where $\vec{r}(t) = a(t)\vec{x}(t)$.

I am trying to derive an equation such that

$$\left.\frac{\partial}{\partial t}\right|_r = \left.\frac{\partial }{\partial t}\right|_x + \left.\frac{\partial \vec{x}}{\partial t}\right|_r \cdot \nabla_x $$

where $\nabla_r = \frac{1}{a}\nabla_x$

It is actually about coordinate transformation.

 

[fresh_42]

Your notation doesn't make any sense. Firstly, you wrote an equation of differential operators, not derivatives. Secondly, you evaluate an operator depending on one variable at a location of another variable. What does that mean? If it is only a coordinate transformation, then you need the chain rule, that's all.