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: 42


$f'(\theta) = -\dfrac{\csc (\sqrt{\theta-1})}{2\sqrt{\theta-1}}$

Work Step by Step

In order to derivate this function you have to apply the chain rule Let's make an «u» substitution to make it easier $u = \sqrt{\theta-1}$ $f(u) = \cot u$ Derivate the function: $f'(u) = -u'\csc u$ Now let's find u' $u' = \dfrac{1}{2\sqrt{\theta-1}}$ Then undo the substitution, simplify and get the answer: $f'(\theta) = -(\dfrac{1}{2\sqrt{\theta-1}})\csc (\sqrt{\theta-1})$ $f'(\theta) = -\dfrac{\csc (\sqrt{\theta-1})}{2\sqrt{\theta-1}}$
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