quick double check on trig function derivatives

Ok so I know the derivative of tanx = sec^2(x) so then would the tangent of tan (x^2) = sec^2(x^2)? or sec^4(x). And then the derivative of that would be ?

I know the derivative of sec(x) = sec(x) tan(x) so sec^2(x) would have derivative sec^2(x)tan^2(x)?

I'm doing derivative of a derivative. I'm normally not so in need of help with math but I ended up in the hospital for a week and kinda feel behind. Thanks for all the help lately on here.

Oh, and the original problem was tan^-1(x^2) so then 1/tan(x^2). Just wondering from there so I can double check my answers.