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: 29


$$y' = {\pi ^{\sin x}}\cos x\left( {\ln \pi } \right)$$

Work Step by Step

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