Question

if x^(p).y^(q) = (x+y)^(p+q) , prove that dy/dx = y/x

Answer 1

Starting with (x^p)*(y^q) = (x+y)^(p+q), take logs of both sides: p.ln(x) + q.ln(y) = (p+q)ln(x+y) differentiate implicitly: p/x + (q/y).dy/dx = (p+q)/(x+y)[(1 + dy/dx)] collect terms in dy/dx: dy/dx[q/y - (p+q)/(x+y)] = (p+q)/(x+y) - p/x dy/dx[(qx+qy-py-qy)/y(x+y)] = (px+qx-px-py)/x(x+y) dy/dx (qx-py)/y = (qx-py)/x dy/dx = y/x