Finding dy/dx of y=1/(x+y) using implicit differentiation: Step-by-step guide

[cj2222]

can someone show me step by step how to find dy/dx of y=1/(x+y) using implicit differentiation?

 

[hunt_mat]

if $$y=1/[x+y)$$ then $$y^{2}+xy=1$$. Differentiate to obtain:
$$2y\frac{dy}{dx}+y+x\frac{dy}{dx}=0$$
Re-arrange to obtain dy/dx

 

[hunt_mat]

Or failing that $$y^{2}+xy-1=0$$ in a quadratic in y, solve this equation and you should have y=y(x) which is easy to differentiate!

 

[Unit]

hunt_mat is correct. Rearranging is much quicker, but taking the quadratic route is very useful to check your answer.

Edit: If I let v = x + y, then y = 1/v and dy/dx = -1/v^2 dv/dx. However, continuing this does not give me the right answer; why not? EDIT: Nevermind, I figured it out. :)