Calculus (3rd Edition)

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

Chapter 3 - Differentiation - 3.6 Trigonometric Functions - Exercises - Page 140: 9


$$ H'(t)=\sec t+2\sin t\sec^2 t\tan t$$

Work Step by Step

Recall the product rule: $(uv)'=u'v+uv'$ Recall that $(\sin x)'=\cos x$. Recall that $(\sec x)'=\sec x\tan x$. Since $ H(t)=\sin t\sec^2t $, then using the product rule, we have \begin{align*} H'(t)&=(\sin t)'\sec^2t+\sin t(\sec^2 t)'\\ &=\cos t\sec^2t+2(\sin t) (\sec t) \sec t \tan t\\ &=\cos t\sec^2t+2\sin t\sec^2 t\tan t\\ &=\sec t+2\sin t\sec^2 t\tan t. \end{align*} (Where we used the fact that $\sec t=\dfrac{1}{\cos t}$ in the last line.)
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