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 146: 73


$$\frac{d^2}{dx^2}\sin (x^2)=2\cos(x^2)-4x^2\sin (x^2).$$

Work Step by Step

Recall that $(\sin x)'=\cos x$. Recall that $(\cos x)'=-\sin x$. Recall that $(x^n)'=nx^{n-1}$ We have $\frac{d}{dx}\sin (x^2)=\cos (x^2) (2x)=2x\cos(x^2)$ and hence $\frac{d^2}{dx^2}\sin (x^2)=2\frac{d}{dx} x\cos(x^2)$ Use the product rule: $=2\cos(x^2)-2x\sin (x^2)(2x)$ $=2\cos(x^2)-4x^2\sin (x^2)$
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