For implicit differentiation or Partial Fluxion as you call it. You apply the following:

1) Take the derivative with respect to x on both sides

2)Any where you see a term with no y's in it, differentiate like normal

3) Now in the terms with y's in them, we differentiate with respect to y but keep the term \frac{dy}{dx}. This is essentially the chain rule.

4) Now in this case we'll have to apply the product rule to the right hand side of the equation

Now we solve for

Now we substitute in the point (1,1) and we get:

So the slope of the tangent line at that point is 0.