Use chain rule
F'(x) = dF(x)/dx = (dF(x)/dx2)(dx2/dx) = 2x (dF(x)/dx2) (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)/dx2 = √(1 + t3) where t = x2 (2)
If you combine (1) and (2) you will obtain your answer:
F'(x) = 2x√(1+x6)
Comment. There is no need to integrate and then differentiate by x.
If you want to type x6 instead of x^6 simply type x6, highlight only 6 and click on the icon (x2).