Differentiate x^cos(x) and find the derivative of cosec^-1(x)

for part a) let y=x^cos(X) , the ln(y)=ln(x^cos(X))=cos(x)ln(x), thus d/dx (ln(y(x)) = d/dx (cos(x)ln(x)), 1/y*dy/dx=cox(x)/x - sinxlnx => solve for dy/dx => y'(x)=x^cos(X) (cox(x)/x - sinxlnx) b) d/dx cosec^-1(x)= -1/x(x² -1)^1/2 this is shown by setting y as the function, rearrange for x then doing implict differentiation to solve for dy/dx in terms of y, then use the defenintions of sine to express in terms of x