Answer: y = x.
- A y = x
- B y = -x + 4
- C y = x + 4
- D y = -x
Correct answer: A. y = x
Explanation: Implicit differentiation of xy = 4: y + xy' = 0, so y' = −y/x = −1 at (2,2). Tangent slope = −1, so normal slope = 1. Normal line: y − 2 = 1·(x − 2), giving y = x.
As point Q slides along the curve toward P, the secant line PQ rotates into the tangent line at P, whose slope is the derivative.
Concept context
Rate of change, differentiation rules, and applications