Calculus, 10th Edition (Anton)

Published by Wiley
ISBN 10: 0-47064-772-8
ISBN 13: 978-0-47064-772-1

Chapter 2 - The Derivative - 2.6 The Chain Rule - Exercises Set 2.6: 58

Answer

$$\frac{dx}{dy}=2\csc^2\left(\frac{\pi}{3}-y\right)\cot\left(\frac{\pi}{3}-y\right)$$

Work Step by Step

$$\frac{dx}{dy}=\frac{d}{dy}\left(\csc^2\left(\frac{\pi}{3}-y\right)\right)=2\csc\left(\frac{\pi}{3}-y\right)\frac{d}{dy}\left(\csc\left(\frac{\pi}{3}-y\right)\right)= 2\csc\left(\frac{\pi}{3}-y\right)\cdot\left(-\csc\left(\frac{\pi}{3}-y\right)\cot\left(\frac{\pi}{3}-y\right)\right)\frac{d}{dy}\left(\frac{\pi}{3}-y\right)=-2\csc^2\left(\frac{\pi}{3}-y\right)\cot\left(\frac{\pi}{3}-y\right)\cdot(-1)=2\csc^2\left(\frac{\pi}{3}-y\right)\cot\left(\frac{\pi}{3}-y\right)$$
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