Calculus, 10th Edition (Anton)

Published by Wiley
ISBN 10: 0-47064-772-8
ISBN 13: 978-0-47064-772-1

Chapter 6 - Exponential, Logarithmic, And Inverse Trigonometric Functions - 6.3 Derivatives Of Inverse Functions; Derivatives And Integrals Involving Exponential Functions - Exercises Set 6.3 - Page 432: 28

Answer

$$y' = - {3^{ - x}}\ln 3$$

Work Step by Step

$$\eqalign{ & f\left( x \right) = {3^{ - x}} \cr & y = {3^{ - x}} \cr & {\text{take logarithm natural on both sides}} \cr & \ln y = \ln {3^{ - x}} \cr & {\text{logarithm properties}} \cr & \ln y = - x\ln 3 \cr & {\text{differentiate}} \cr & \left( {\ln y} \right)' = \left( { - x\ln 3} \right)' \cr & \left( {\ln y} \right)' = - \ln 3\left( x \right)' \cr & \frac{{y'}}{y} = - \ln 3 \cr & y' = y\left( { - \ln 3} \right) \cr & {\text{replace }}y = {3^{ - x}} \cr & y' = {3^{ - x}}\left( { - \ln 3} \right) \cr & {\text{simplify}} \cr & y' = - {3^{ - x}}\ln 3 \cr} $$
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