Given : xp * yq =(x+y)p+q
Take log on both side
log xp + log yq = log(x+y)p+q
=> plog x + qlogy = (p+q)*log (x+y)
Now differentiate with respect to x
p/x + (q/y)* dy/dx = {(p+q)/(x + y)}*(1 + dy/dx)
=> p/x + (q/y)* dy/dx = {(p+q)/(x + y) + {(p+q)/(x + y)}*(dy/dx)
=> (q/y)* dy/dx - {(p+q)/(x + y)}*(dy/dx) = (p+q)/(x + y) - p/x
=> {(q/y) - (p+q)/(x + y)}*(dy/dx) = (p+q)/(x + y) - p/x
=> [(qx + qy - py - qy)/{y*(x + y)}]*(dy/dx) = (px + qx - px - py)/{x*(x+y)}
=> {(qx - py)/y}*(dy/dx) = (qx - py)/x
=> (dy/dx)/y = 1/x
=> dy/dx = y/x