Remember that the curvature is given by k=dQ/ds, where Q is the angle between the tangent vector (x'(t),y'(t)) and (1,0), and ds the arclength element. Begin by

k=dQ/ds=(dQ/dt)/(ds/dt),

and use:

-> (ds/dt)^2= (x')^2+(y')^2

but also

-> Q=Arctan(dy/dx)=Arctan([dy/dt]/[dx/dt]),

etc.