Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 2 - Derivatives - 2.5 The Chain Rule - 2.5 Exercises: 54

Answer

$$y=-x+\pi/2$$

Work Step by Step

First we will find the derivative: $$y'=(\sin^2x\cos x)'=(\sin^2x)'\cos x+\sin^2x(\cos x)'= 2\sin x(\sin x)'\cos x+\sin^2x(-\sin x)=2\sin x\cos x\cdot \cos x-\sin^3x=\sin x(2\cos^2x-\sin^2x)=\sin x(\cos^2x+\cos 2x)$$The tangent line of the curve at the point $(x_{0},y_{0})$ is given by formula: $$y-y_{0}=y'(x_{0})(x-x_{0}).$$ For $(x_{0},y_{0})=(\pi/2,0)$ we get: $$y-0=\left.\sin x_{0}(\cos^2x_{0}+\cos2x_{0})\right|_{x_{0}=\pi/2}(x-\pi/2)\Rightarrow y=\sin\pi/2(\cos^2\pi/2+\cos2\pi/2)(x-\pi/2)\Rightarrow y=1\cdot(0+(-1))(x-\pi/2)\Rightarrow y=-x+\pi/2$$
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