Just to clarify what you are asking, when you say you want the derivatives of sec^-1(x) and csc^-1(x), do you mean arcsec(x) and arccsc(x), or the reciprocals of secant and cosecant?

Because sec^-1(x) = cos(x) and csc^-1(x) = sin(x), so it's not really clear to me which you mean. 

Yes, I apologize for the error. I did mean arcsec(x) and arccsc(x). Thats my bad

These videos worked for me, since I don't think that KA has proofs for more advanced trig derivatives like those. Feel free to check these out, and let me know if they help. If not, I can try to find additional resources. 

arcsec(x): https://www.youtube.com/watch?v=VjNWN6j_Pgs

arccsc(x): https://www.youtube.com/watch?v=WFCWtycjKAk

Your lucky my biggest weakness is Colby Cartier gave alow I am failure with the trigonometry stuff, I am self teaching myself maths and science since I started Khan academy I have done and understood maths more than ever now I am 38 yr old and wanting to self teach myself and if something comes out of it good if not well, anyway I can not imagine how far iv came since I started using this site, I am from the uk this site may be American but it has helped me a lot, now maybe in time I can get into physics and engineering I just simply love maths,and I have a minor learning disability and I can not imagine that I was crap at maths once before, maybe now I can pick  my borderline grade up when the time comes ill be ready, I mean alow I understand the trig now I still get confused with the trig type sum Colby wrote.

I have never wrote an essay or thesis in my life I would love to practice that and them somehow, but I did used to write for my school paper and tried to write novels and stuff like that unlucky I could never keep it going after chapter 1 iv not got a good attention span and I may have an over active imagination but I can never seem to keep it going.

Thank you Ayesha, I will look into those videos.