Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 7: Transcendental Functions - Section 7.3 - Exponential Functions - Exercises 7.3 - Page 391: 56

Answer

$$\frac{{dy}}{{dx}} = {3^{ - x}}\ln \left( {\frac{1}{3}} \right)$$

Work Step by Step

$$\eqalign{ & y = {3^{ - x}} \cr & {\text{Find the derivative of }}y{\text{ with respect to }}x \cr & \frac{{dy}}{{dx}} = \frac{d}{{dx}}\left[ {{3^{ - x}}} \right] \cr & {\text{Use the rule }}\frac{d}{{dx}}\left[ {{a^u}} \right] = {a^u}\ln a\frac{{du}}{{dx}}. {\text{Let }}a = 3{\text{ and }}u = - x{\text{,}} \cr & \frac{{dy}}{{dx}} = {3^{ - x}}\ln 3\frac{d}{{dx}}\left[ { - x} \right] \cr & {\text{Solve the derivative and simplify}} \cr & \frac{{dy}}{{dx}} = {3^{ - x}}\ln 3\left( { - 1} \right) \cr & \frac{{dy}}{{dx}} = - {3^{ - x}}\ln 3 \cr & \frac{{dy}}{{dx}} = {3^{ - x}}\ln \left( {\frac{1}{3}} \right) \cr} $$
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