Calculus 10th Edition

Published by Brooks Cole
ISBN 10: 1-28505-709-0
ISBN 13: 978-1-28505-709-5

Chapter 2 - Differentiation - 2.5 Exercises: 12

Answer

$\dfrac{dy}{dx}=\dfrac{\cos{\pi x}}{\sin{\pi y}}.$

Work Step by Step

$\dfrac{d}{dx}((\sin{\pi x}+\cos{\pi y})^2)=\dfrac{d}{dx}(2)\rightarrow$ Using the Chain Rule with $u=\sin{\pi x}+\cos{\pi y}\rightarrow\dfrac{du}{dx}=\pi\cos{\pi x}-\pi\sin{\pi y}\rightarrow$ $\dfrac{d}{dx}((\sin{\pi x}+\cos{\pi y})^2)$ $=2(\sin{\pi x}+\cos{\pi y})(\pi\cos{\pi x}-\dfrac{dy}{dx}(\pi\sin{\pi y)})$ $2(\sin{\pi x}+\cos{\pi y})(\pi\cos{\pi x}-\dfrac{dy}{dx}(\pi\sin{\pi y)})=0\rightarrow$ $\dfrac{dy}{dx}(\pi\sin{\pi y})=\pi\cos{\pi x}\rightarrow$ $\dfrac{dy}{dx}=\dfrac{\cos{\pi x}}{\sin{\pi y}}.$
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