$x’(t)=\frac {dx} {dt}$
$y’(t)=\frac {dy} {dt}$
$\frac {dy} {dx} =\frac {\frac {dy} {dt}}{\frac {dx} {dt}}$
$\frac {dy} {dx} =\frac {\frac {dy} {\theta}}{\frac {dx} {d\theta}}=\frac {rcos\theta+r'sin\theta}{-rsin\theta+r'cos\theta}, \frac {dx} {d\theta}\ne0$
$\frac {d^2y} {dx^2} ={\frac d {dx}[\frac {dy} {dx}]}=\frac d {dt}[\frac {dy} {dx}]\cdot\frac {dt}{dx}$
$L=\int_a^b\sqrt{(\frac {dx} {dt})^2+(\frac {dy} {dt})^2}$
$|\vec v(t)|=||\vec v(t)||=\sqrt{(\frac {dx} {dt})^2+(\frac {dy} {dt})^2}$
$\vec s =\lang x(t), y(t)\rang$
$\vec v =\lang x'(t), y'(t)\rang$