The derivative of the inverse sine of x with respect to x is 1/sqrt(1-x^2), where "sqrt" stands for square root. Inverse sine of x is sometimes written as arcsin(x). By definition, arcsin(x) = -iln(ix + sqrt(1-x^2)), where i is the square root of -1 and ln is the natural logarithm.
One can use the chain rule of differentiation to compute the derivative of inverse sine. The chain rule says that dy/dx = dy/du du/dx. Letting y = arcsin(x) = -iln(ix + sqrt(1-x^2)) = -iln(u) and u = ix + sqrt(1-x^2) gives the result dy/du = -i/u and du/dx = i - x/sqrt(1-x^2). Substituting these results into the chain rule gives d(arcsin(x))/dx = 1/sqrt(1-x^2).