I want to construct a frenet curve $Y: \Bbb R \to \Bbb R^3$ with constant curvature $K$ and torsion $T$. I figured I'd start with calculating the derivatives of $K$ and $T$ but I don't know how to differentiate those expressions. They are specifically,
$$K = \|(Y' \times Y'')\|/\|Y'\|^3,$$
$$T = \det[Y', Y'', Y''']/\|Y' \times Y''\|^2.$$
Thanks for your time!
You might want to consider exploiting the existence and uniqueness theorem for such curves, given $K$ and $T$. By the theorem, it suffices to exhibit a curve with the desired properties, and every other curve will have to be congruent to the one exhibited. You could start with the helix $\alpha(t) = (a \cos \omega t, a \sin \omega t, bt)$.
The Frenet-Serret equations and the fundamental theorem of the theory of curves can be found in most introductory textbooks in classical differential geometry. For example you can consult the book by do Carmo "differential geometry of curves and surfaces" where the proof is given on pages 309-311.