${x^2} + xy + {y^2} = 100$

We are given that, ${x^2} + xy + {y^2} = 100$

...(i)
Differentiating (i) on both sides w.r.t. x, we get

$2x + x\cfrac{{dy}}{{dx}} + y + 2y\cfrac{{dy}}{{dx}} = 0$ $\Rightarrow$ $\cfrac{{dy}}{{dx}}[x + 2y] = - (2x + y)$

$\Rightarrow$ $\cfrac{{dy}}{{dx}} = \cfrac{{ - (2x + y)}}{{(x + 2y)}}$