Showing a parametric equation is smoothly parametrized?

A parametric equation, say *r(t)*, is smoothly parametrized if:

1. its derivative is continuous, and

2. its derivative does not equal zero for all *t* in the domain of *r*.

Now that sounds simple enough. Now lets say we have the tractrix:

*r(t) = (t-tanht)i + sechtj*,

then *r'(t) = [ 1-1/(1+t^2) ]i + [ tantsect ]j*, right?

FIRSTLY, do I have the derivative correct? --and

SECONDLY, without reverting to MatLab or Maple to view the graph, how do we deduce that *r'(t)* is continuous (or not)?

Do I just state that is is/isn't -by inspection, or ...?