Note in the following explanation . I prefer using instead of . Old habits die hard...
Implicit differentiation yields:
(remember you have to use product rule...x and y are both variables...so when differentiating the product we have to use product rule)
We want all the terms with on one side so we can factor out that and solve for it. So the previous equation turns into:
Factoring out on the left hand side yields:
Now, to get by itself, we must divide through by which results in:
You want to find the slope of the tangent line at (x,y)=(2,1). We have solved for which gives the slope of the tangent line at any point on the original function. So now, you just need to substitute x=2 and y=1 into the equation for .
Does that make sense?