University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 3 - Section 3.6 - The Chain Rule - Exercises - Page 158: 78



Work Step by Step

$$y=\sin(x^2e^x)$$ - Find $y'$: $$y'=\cos(x^2e^x)(x^2e^x)'=\cos(x^2e^x)(2xe^x+x^2e^x)$$ - Find $y''$: $$y''=\Big(\cos(x^2e^x)\Big)'(2xe^x+x^2e^x)+\cos(x^2e^x)(2xe^x+x^2e^x)'$$ $$y''=-\sin(x^2e^x)(x^2e^x)'(2xe^x+x^2e^x)+\cos(x^2e^x)\Big(2(xe^x)'+(x^2e^x)'\Big)$$ $$y''=-\sin(x^2e^x)(2xe^x+x^2e^x)(2xe^x+x^2e^x)+\cos(x^2e^x)(2e^x+2xe^x+2xe^x+x^2e^x)$$ $$y''=-\sin(x^2e^x)(2xe^x+x^2e^x)^2+\cos(x^2e^x)(2e^x+4xe^x+x^2e^x)$$ $$y''=e^x\cos(x^2e^x)(2+4x+x^2)-\sin(x^2e^x)(2xe^x+x^2e^x)^2$$
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