Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 3 - Differentiation - 3.7 The Chain Rule - Exercises - Page 145: 4

Answer

See the attachment.

Work Step by Step

We are given: $f(u)=u^{4}+u$, $g(x)=\cos x$ The composite function is: $f(g(x))=(\cos x)^{4}+\cos x=\cos^{4}x+\cos x$ Take the derivatives: $f'(u)=4u^{3}+1$ $f'(g(x))=4(\cos x)^{3}+1= 4\cos^{3}x+1$ $g'(x)=-\sin x$ $(f \circ g)'=f'(g(x))g'(x)$ $= (4\cos^{3}x+1)\times -\sin x$. $=-\sin x(4\cos^{3}x+1)$ See the filled table below.
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