Given: $F(x,y,z)=x^2+3xy+2yz+y^2+z^2-11=0$.
Does $F$ implicitly define a function $z = f(x,y)$ around the point $(x_0,y_0,z_0)=(1,2,0)$.
If so determine $f_x=\frac{df}{dx}=\frac{dz}{dx}$ and $f_y=\frac{df}{dy}=\frac{dz}{dy}$ by the implicit-function theorem and evaluate them at the given point.
What I have done so far: solving for $dz/dx$ and $dz/dy$
$$\frac{dz}{dx}= - (2x+y)/(2y+z)$$ $$\frac{dz}{dy}= (x-2z-2y)/(2y+2z)$$
but I have no clue how to evaluate this.
Why don't you just plug the values of the variables into your calculated formulas?
BTW you did not prove properly that you can use the formulas - there is some (boring but fast) work to be done there. Hint: something about some derivative matrix being invertible.