To find f'(x) use the product rule for the 1st term and power-rule/chain-rule for 2nd term,
f'(x) = x (-1)/(√(1-x2) ) + cos-1(x) -1/2 (1 - x2)-1/2 (-2x)
To find f'(x) at √3/2 we could substitute right now, but chances are the formula for f'(x) will simplify significantly.
f'(x) = cos-1(x) - x/√(1-x2) + x/√(1 - x2) = cos-1(x)
So f'(√3/2) = cos-1(√3/2) = ?? (what angle has a cos of √3/2?)