Originally Posted by

**kodos** Let F: $\displaystyle R^2 \rightarrow R$ be a function $\displaystyle C^2$ for which F(x+y,y+z)=0 defines implicitly z=f(x,y) around a point (a,b,c). Verify that in (a,b)

dz/dx-dz/dy=1

I'm having difficulty with getting rid of some derivatives, they should cancel out I guess.

(*It's a bit heavy to read but please take a look at my work)*

First, I apply the chain rule with u=g(x,y)=x+y, v=h(x,y)

and get the expressions for dz/dx and dz/dy

dF/dx=dF/du dg/dx+dF/dv dh/dx=dF/du+dF/dv dz/dx

solve for dz/dx,

dz/dx=(dF/dx-dF/du)/(dF/dv)

Then,

dF/dy=dF/du+dF/du dz/dy

dz/dy={dF/dy-dF/du}/{dF/dv}

So dz/dx-dz/dy={dF/dx-dF/dy-2dF/du}/{dF/dv}=???

From dF/dz=dF/dv but dF/dz is no good.

How can I get rid of those derivatives?