## Differential arc of a trajectory

The trajectory of a particle is given by $\vec r(t)$ and its velocity by $\vec v(t)$.
Calculate $d\vec s = dx \hat i + dy \hat j + dz \hat k$ and $ds^2$ in cylindrical and spherical coordinates.
So far I have converted $\hat i$, $\hat j$ and $\hat k$ into a combination of unit vectors in cylindrical coordinates. I also done the opposite.
I just don't know how to convert dx into a differential depending on $r$, $\varphi$ and z. I realize it will only depend on $r$ and $\varphi$.
I know that $x=r\cos \varphi$. I'm not really sure how to get dx. I don't think I can call "dx" a total derivative.