If f(x)=1/sin(x), then f(x)=csc(x) and arccsc(f(x))=x.
dx/df is the derivative of arccsc(f(x)).
Let y=f(x)=1/sin(x), dy/dx=-cos(x)/sin²(x)=-1/(tan(x)sin(x)) or -cot(x)csc(x).
dx/df=dx/dy=1/(dy/dx)=-tan(x)sin(x)=-sin²(x)/cos(x)=-(1/y²)/√(1-(1/y²)).
dx/df=-(1/y²)/√(1-(1/y²))=-(1/y²)(y/√(y²-1)=-1/(y√(y²-1)).
Therefore the derivative of arccsc(f(x))=-1/[f(x)√((f(x))²-1)].