coordinate transformation

I have already solved for the scaling factors h1,h2,h3 and the corresponding unit vectors. I need to find the position vector for the coordinate system I am working with. I am looking over the position vector for spherical and cylindrical and it seems straight forward. The formula for dR= h1*du1*u1Hat + h2*du2*u2hat + h3 *du3*u3hat. It is pretty easy to see how for spherical the chain rule and the derivative of the rhat term gives the other pieces. Is there a method to doing this for something that is not so obvious and the derivatives not so straight forward like in the case of prolate spherical cordinates.

Thanks