Calculus: Early Transcendentals (2nd Edition)

Published by Pearson
ISBN 10: 0321947347
ISBN 13: 978-0-32194-734-5

Chapter 3 - Derivatives - 3.10 Derivatives of Inverse Trigonometric Functions - 3.10 Execises - Page 222: 42

Answer

$$ - \frac{1}{8}$$

Work Step by Step

$$\eqalign{ & {\text{Let }}f\left( x \right) = {x^2} - 2x - 3;\,\,\,\,\,\left( {12, - 3} \right) \cr & {\text{Calculate }}f'\left( { - 3} \right) \cr & f'\left( x \right) = 2x - 2 \cr & f'\left( { - 3} \right) = 2\left( { - 3} \right) - 2 \cr & 3f'\left( { - 3} \right) = - 8 \cr & {\text{Find the derivative of the inverse function}}{\text{, using the THEOREM 3}}{\text{.23}} \cr & \left( {{f^{ - 1}}} \right)'\left( {{y_0}} \right) = \frac{1}{{f'\left( {{x_0}} \right)}},\,\,\,\,{\text{Where }}{y_0} = f\left( {{x_0}} \right) \cr & {\text{Then}}{\text{,}} \cr & \left( {{f^{ - 1}}} \right)'\left( {12} \right) = \frac{1}{{ - 8}} \cr & \left( {{f^{ - 1}}} \right)'\left( {12} \right) = - \frac{1}{8} \cr} $$
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