Use chain rule
F'(x) = dF(x)/dx = (dF(x)/dx^{2})(dx^{2}/dx) = 2x (dF(x)/dx^{2}) (1)
When you differentiate the integral with variable upper limit it gives you the function being integrated.
Thus (because the lower limit doesn't play any role being a constant number)
dF(x)/dx^{2} = √(1 + t^{3}) where t = x^{2 (2)}
If you combine (1) and (2) you will obtain your answer:
F'(x) = 2x√(1+x^{6})
Comment. There is no need to integrate and then differentiate by x.
If you want to type x^{6} instead of x^6 simply type x6, highlight only 6 and click on the icon (x^{2}).
Good luck!