Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 3 - Section 3.3 - Derivatives of Trigonometric Functions - 3.3 Exercises - Page 196: 18


1) Rewrite $\sec x$ into $\frac{1}{cos x}$ 2) Then apply the Quotient Rule for $\frac{d}{dx}(\frac{1}{cos x})$ 3) Use trigonometrical rules to simplify.

Work Step by Step

We know that $\sec x=\frac{1}{\cos x}$. So, $$\frac{d}{dx}(\sec x)=\frac{d}{dx}\Bigg(\frac{1}{\cos x}\Bigg)$$ Now apply the Quotient Rule, we have $$\frac{d}{dx}(\sec x)=\frac{(1)'\cos x-1(\cos x)'}{(\cos x)^2}$$ $$\frac{d}{dx}(\sec x)=\frac{0\times\cos x-(-\sin x)}{\cos^2 x}$$ $$\frac{d}{dx}(\sec x)=\frac{\sin x}{\cos^2 x}$$ $$\frac{d}{dx}(\sec x)=\frac{\sin x}{\cos x}\times\frac{1}{\cos x}$$ $$\frac{d}{dx}(\sec x)=\tan x\sec x$$ The statement has been proved.
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