Let's say I have a change of variables $(x,y) \to (w,z)$. I would like to express $\nabla f(x,y)= \partial_x f \hat{x} + \partial_y f \hat{y}$ in the other coordinates ($\hat{x}$ and others are vectors of norm 1). That means I should find something like: $$ \nabla f(w,z)= A(w,z) \partial_w f \hat{w} + B(w,z)\partial_z f \hat{z}$$
If you use polar coordinates you can guess who they are from a geometrical perspective, but in general do you have just to write everything as a function of $(w,z)$, do some matrix products and take things outside multiplicating until its norm gets 1? For instance, let's take the transformation $(x,y) \to (w,z)$ where $w=x^3$ and $z=y$ (let's work on $\mathbb{R}^+$). Then I would get $$ \nabla f= 3w^{2/3} \partial_w f \hat{w}+ \partial_z f \hat{z}$$
Where the vectors are the standard base of $\mathbb{R}^2$. But I have found that only after deriving etc. (in this case it was also easy though). Is there a way to tell in general who those vectors will be without doing all the work?