(Just a heads up, I'm going to use arccos(x) to denote inverse cosine here.)
This can be a tricky sort of problem, much like differentiating x^x. Whenever there is a variable in the base AND the exponent of a function, we must use logarithmic differentiation. If that doesn't ring any bells, basically we take the log of both sides, do implicit differentiation, and solve for y'.
Take ln[ ] of both sides:
ln(y) = ln[x^arccos(x)].
Use logarithm power rule:
ln(y) = arccos(x)*ln(x).
Differentiate (implicit diff on the left, product rule on the right):
(1/y)*(y') = arccos(x)*(1/x) + (-1/√[1-x2])*ln(x)
y'/y = arccos(x)/x - ln(x)/√[1-x2].
Solve for y':
y' = y*[ arccos(x)/x - ln(x)/√[1-x2] ]
Substitute original function in for y:
y' = [x^arccos(x)]*[ arccos(x)/x - ln(x)/√[1-x2] ]