If `x^m y^n=(x+y)^(m+n),prove (dy)/(dx)=y/xdot`

Explanation

`"We have "x^(m)y^(n)=(x+y)^(m+n).`
Taking log on both sides, we get m log x + n log y = (m+n) log (x+y)
Differentiating both sides w.r.t.x, we get
`m(1)/(x)+n(1)/(y)(dy)/(dx)=(m+n d)/(x+y dx)(x+y)`
`"or "(m)/(x)+(n)/(y)(dy)/(dx)=(m+n)/(x+y)(1+(dy)/(dx))`
`"or "{(n)/(y)-(m+n)/(x+y)}(dy)/(dx)=(m+n)/(x+y)-(m)/(x)`
`"or "{(nx+ny-my-ny)/(y(x+y))}(dy)/(dx)={(mx+nx-mx-my)/((x+y))}`
`"or "(nx-my)/(y(x+y))(dy)/(dx)=(nx-my)/((x+y)x)`
`"or "(dy)/(dx)=(y)/(x)`