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A boat is at a distance A from the dock and is moored at point O by a rope of length A. A girl loosens the mooring and walks along the quayside as she pulls the boat after her with the rope, which is constantly tight. The boat's movement follows the dotted curve on the figure. The rope is constantly tangent to this curve

What is the derivative to the function?

task

I know that this curve is called a "Tractrix" and I do find the formula for the derivative at wikipedia. But I don't understand how they find the derivative. I know it is something about a right angle triangle. Could someone explain it in a simple way for me? And how do I find the function using the derivative?

Mathomat55
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1 Answers1

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From the diagram, we have

$$\cos\theta = \frac xa\tag 1$$

and the derivative of the function is $$y'=-\tan\theta = -\sqrt{\sec^2\theta -1}= -\sqrt{\frac1{\cos^2\theta} -1}\tag 2$$

Plus (1) into (2) to obtain the derivative

$$y'= -\frac{\sqrt{a^2-x^2}}x$$

Quanto
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  • Thank you for the answer! But how do you see that the derivative is tangent? And where does the minus sign come from? – Mathomat55 Feb 07 '20 at 21:27
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    @Mathomat55 - The slope of the tangent line, which is also its derivative, is defined as $\tan\alpha$, where $\alpha$ is the angle the tangent line with respect to the $x$-axis. In your problem, $\alpha = 180 - \theta$. Thus, $y'=\tan\alpha=-\tan\theta$. – Quanto Feb 07 '20 at 21:30