Let's say I have the gradient in 3 dimensions and want to approximate it using finite differences: $$ \nabla_{\vec{x}}f(\vec{x}) \approx \sum_i \frac{f(\vec{x}+\vec{e}_i\Delta x_i)-f(\vec{x})}{\Delta x_i} \vec{e}_i $$
where $\vec{e}_i$ are the cartesian unit vectors. Now I want to make a change of basis. The new basis $\mathcal{B}=\left\{\vec{a}_1,\vec{a}_2,\vec{a}_3\right\}$ is connected to the old one via
$$ \vec{x} = A \cdot \vec{y} $$
for some matrix $A$ connected to $\mathcal{B}$. Say I have some function $g(\vec{x})$ that lives on discrete points that are easily described by the new basis and only hard to describe using the old basis (for example: $\vec{P}=\sum_i\delta_i \vec{a}_i$ for all points). I can rewrite
$$ g(\vec{x}) = g(A\vec{y}) = \tilde{g}(\vec{y})$$
Then, I can approximate
$$ \nabla_{\vec{y}}\tilde{g}(\vec{y}) \approx \sum_i \frac{\tilde{g}(\vec{y}+\frac{\vec{a}_i}{\left|\vec{a}_i\right|}\Delta y_i)-\tilde{g}(\vec{y})}{\Delta y_i} \frac{\vec{a}_i}{\left|\vec{a}_i\right|} $$
(as I was typing this I was not sure if I need the normalization of $\vec{a}_i$...)
Finally, I can formulate my question. Suppose I need $\nabla_{\vec{x}}g(\vec{x})$ for some expression. How do I have to replace it by
$$ \sum_i \frac{\tilde{g}(\vec{y}+\frac{\vec{a}_i}{\left|\vec{a}_i\right|}\Delta y_i)-\tilde{g}(\vec{y})}{\Delta y_i} \frac{\vec{a}_i}{\left|\vec{a}_i\right|} $$
i.e. the approximation for $\nabla_{\vec{y}}\tilde{g}(\vec{y})$. How are these expresions related? I assume that the Jacobian appears somewhere... For some reasons I cannot wrap my head around this.