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The acceleration however is tedious by my current method and I'm wondering if there is some shortcut or superior method I'm not aware of.

My current method is to take the time derivative of velocity, which includes taking the time derivative of the basis vectors. This is the standard approach I've seen in mechanics texts for spherical and cylindrical. Is there any other way to find acceleration? We have looked at the metric tensor, but is there a way to get acceleration from it perhaps?