I'm having two sets of grids. One is uniform and another one is not uniform. I will calculate the derivative in uniform grid points and I like to transfer(map) the derivative to the non-uniform grid points. I have the relationship between uniform and non-uniform grid points.

A similar question was already asked. I'd like to get one answer with an example. For the time sake, I consider the following simple example.

The relation between my uniform and the non-uniform grid is $$x= \zeta^2$$ Let $y(x)$ be the function for which I'd like to calculate the derivative on the non-uniform grid and $y = x^2$. So, $\frac{dy}{dx} =2x$

The uniform grid and derivative information are given, as follows:

The non-uniform grid: .

I want to find the derivative information in the non-uniform grid by mapping.