y=x^siny
take logs of both sides
ln(y) = siny.ln(x)
now differntiate, wrt y, and use the product rule on the rhs
(1/y) = (cosy)*ln(x) + siny*(1/x)*(dx/dy)
1/y - cosy.ln(x) = (siny)/x*dx/dy
(1 - y.cosy.ln(x))/y = (siny)/x*dx/dy
dy/dx = y.siny/(x{1 - y.cosy.ln(x)})