Why are the r dot terms (eg -r^dotsincos in the z derivative) in the derivatives of the spherical coordinates? Differentiation, as I've understood it, is differentiating a function with respect to a variable, ie time. How can such a variable be shared by both the angles between a point and the axes and the distance between that point and the axis?
I've elaborated on where the question arises in my thought process as I'd like to iron out any misconceptions of differentiation I have while you guys are at it.
At any point in time, the particle's position is described by a set of spherical coordinates. Unless the time dependence of the particle's position is somehow restricted, in the general case all three spherical coordinates will change over time. If e.g. $r$ didn't depend on time, the particle would be restricted to the surface of a sphere of radius $r$.