Calculus, 10th Edition (Anton)

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

Chapter 2 - The Derivative - 2.5 Derivatives of Trigonometric Functions - Exercises Set 2.5: 23

Answer

$\frac{d^2y}{dx^2}=-4\sin x\cos x$

Work Step by Step

$y=sinxcosx$ $\frac{dy}{dx}=(cosx)\times{(cosx)}+(sinx)\times{(-sinx)}$ $\frac{dy}{dx}=cos^2x-sin^2x$ First write it as $\frac{dy}{dx}=(cosx)^2-(sinx)^2$ to spot the chain rule easier. $\frac{d^2y}{dx^2}=2(cosx)^{2-1}\times\frac{d}{dx}(cosx)-(2(sinx)^{2-1}\times\frac{d}{dx}(sinx))$ $\frac{d^2y}{dx^2}=2(cosx)^1\times{(-sinx)}-(2(sinx)^{1}\times{(cosx)})$ $\frac{d^2y}{dx^2}=-2sinxcosx-2sinxcosx$ $\frac{d^2y}{dx^2}=-4\sin x\cos x$
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