Given the following question:
Consider the curves C1 and C2 defined as follows;
C1 : xy = 4 , x > 0
C2 : y^2 - x^2 = 2 , x > 0
Let P(a,b) be a unique point where the curves C1 and C2 intersect. Show that the tangent C1 at P is perpendicular to the tangent to C2 at P.
I used implict differentiation to find the derivatives of both curves, with dy/dx = -y/x and dy/dx = x/y.
The solution provided shows multiplying the two derivatives together, netting a result of -1. Then "Therefore, the result follows".
Why is multiplying the derivatives the correct process and where do I take the answer to net the solution?