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: 34


$f'(x) = \dfrac{3(\sqrt{x+1} -1)^{1/2}}{4\sqrt{x+1}}$

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 $u = \sqrt{x+1} -1$ $f(u) = u^{3/2}$ Derivate the function: $f'(u) = \dfrac{3}{2}u^{1/2}u'$ Now let's find u' $u' = \dfrac{1}{2\sqrt{x+1}}$ Then undo the substitution, simplify and get the answer: $f'(x) = (\dfrac{3}{2}(\sqrt{x+1} -1)^{1/2})(\dfrac{1}{2\sqrt{x+1}})$ $f'(x) = \dfrac{3(\sqrt{x+1} -1)^{1/2}}{4\sqrt{x+1}}$
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