If $\lambda, \mu, \nu$ are the roots of the equation in $k$, $$\frac{x}{a+k}+ \frac{y}{b+k}+\frac{z}{c+k} = 1,$$ then prove that $$\frac{\partial(x,y,z)}{\partial(\lambda, \mu, \nu)} = - \frac{(\mu-\nu)(\nu-\lambda)(\lambda-\mu)}{(b-c)(c-a)(a-b)}$$
I have tried it enough by finding the partial derivatives but the calculations are going out of my reach. Is there any simple way of doing this?