The equation of a tangent plane at is:
We need to find and , but because z is defined implicitly, we need to be careful finding these partial derivatives.
First, let's keep y constant and differentiate implicitly with respect to x to find (or ):
(Be careful differentiating the first term of the given equation - you must use the product rule!)
Rearranging to solve for the partial derivative, you get:
Therefore, at P(1,1,-1), you get:
Similarly, for (or ), we get:
Rearranging:
At P(1,1,-1):
Now, plugging these values into the tangent plane formula, we get:
Making things nice and beautiful, the final answer is: