Chapter 7.P, Problem 6P

A man initially standing at the point O walks along a pier pulling a rowboat by a rope of length L. The man keeps the rope straight and taut. The path followed by the boat is a curve called a tractrix and it has the property that the rope is always tangent to the curve (see the figure).

(a) Show that if the path followed by the boat is the graph of the function $y=f\left(x\right)$ , then

$f\prime \left(x\right)=\frac{dy}{dx}=\frac{-\sqrt{L^2-x^2}}{x}$

(b) Determine the function $y=f\left(x\right)$ .