Use implicit differentiation to find the slope of the tangent line to the curve?

Implicit differentiation means that if you could solve an equation for y, it would equal some function of x.  So, every time you see a y, pretend that it is "hiding" a function of x.  Thus, you will need to use the chain rule on it.  

So, the derivative is:
6x -(3y+3x(dy/dx)) - 9y^2(dy/dx) = 0
notice, I had to use the product rule on 3xy

Solve for dy/dx:
6x - 3y - 3x(dy/dx) - 9y^2(dy/dx) = 0
-3x(dy/dx) - 9y^2(dy/dx) = 3y - 6x
factor out dy/dx:
(dy/dx)(-3x - 9y^2) = 3y - 6x
so,
dy/dx = (3y - 6x)/(-3x - 9y^2)

Now plug in your point where x=1 and y=1
dy/dx at (1,1) = (3 - 6)/(-3 - 9) = -3/-12 = 4


Hope this helped.