kill(all)$

van=facebook.com/LearningCalculus$

y(t):=sqrt(t^2+t-a*sqrt(t));

x(t):=sqrt(sqrt(t)/a-t^2-t);

red(t):=[t, y(t)]$ red(t);

blu(t):=[''(sqrt(sqrt(y(t))/a-y(t)^2-y(t))),''(y(t))];

      dy/dx red = dy/dt / dx/dt red
      dy/dx blu = dy/dt / dx/dt blu

dy_dx_red(t) := diff(red(t)[2],t) / diff(red(t)[1],t)$ dy_dx_red(t);

dy_dx_blu(t) := fullratsimp(diff(blu(t)[2],t) / diff(blu(t)[1],t))$ dy_dx_blu(t);