Differentiate both sides of the equation with respect to xx:
d/dx(ysqrtx-xsqrty) = 0ddx(y√x−x√y)=0
d/dx(ysqrtx) = d/dx(xsqrty) ddx(y√x)=ddx(x√y)
using the product rule:
y/(2sqrtx)+ dy/dx sqrtx = x/(2sqrty)dy/dx + sqrtyy2√x+dydx√x=x2√ydydx+√y
solving for dy/dxdydx:
dy/dx(sqrtx-x/(2sqrty)) = sqrty-y/(2sqrtx)dydx(√x−x2√y)=√y−y2√x
dy/dx((2sqrtxsqrty-x)/(2sqrty)) = (2sqrtxsqrty-y)/(2sqrtx)dydx(2√x√y−x2√y)=2√x√y−y2√x
dy/dx = ((2sqrtxsqrty-y)/(2sqrtx))((2sqrty)/(2sqrtxsqrty-x))dydx=(2√x√y−y2√x)(2√y2√x√y−x)
dy/dx = ((2ysqrtx-y)/(2xsqrty-x))dydx=(2y√x−y2x√y−x)
dy/dx = (y(2sqrtx-1))/(x(2sqrty-1))dydx=y(2√x−1)x(2√y−1)
