It's just a mega-chain rule deal. I'm not sure what help there is, other than demonstrating it. Once you see it, you'll have it firmly in your mind and never trip over another one.

I added the final "(1)" only to emphasize that the argument of the secant also needed some attention.

I like to write these out in a thoretical sense. It helps build understanding of the phases of the solution.

If z(t) = h(g(r(t))), then z'(t) = h'(g(r(t)))*g'(r(t))*r'(t).

In your case, you have h(t) = t^6, g(t) = sec(t), and r(t) = t+1.