Let y = (sin x)^y

Take log on both side, we get

=> log y = log(sin x)^y

=> log y = y*log(sin x)

=> (1/y)*(dy/dx) = (dy/dx)*log(sin x) + (y/sin x)*cos x

=> (1/y)*(dy/dx) - (dy/dx)*log(sin x) = (y*cos x)/sin x

=> {(1/y) - log(sin x)}*(dy/dx) = (y*cos x)/sin x

=> dy/dx = {y*cos x/sin x}/{(1/y) - log(sin x)}

=> dy/dx = {y*cos x/sin x}/{1 - ylog(sin x)/y}

=> dy/dx = {y² *cos x/sin x}/{1 - ylog(sin x)}

=> dy/dx = {{(sin x)^y }² *cot x}/{1 - (sin x)^y *log(sin x)}