Calculus with Applications (10th Edition)

Published by Pearson
ISBN 10: 0321749006
ISBN 13: 978-0-32174-900-0

Chapter 13 - The Trigonometric Functions - 13.2 Derivatives of Trigonometric Functions - 13.2 Exercises - Page 688: 13


$$\frac{{dy}}{{dx}} = 4{e^{4x}}\cos {e^{4x}}$$

Work Step by Step

$$\eqalign{ & y = \sin {e^{4x}} \cr & {\text{differentiate with respect to }}x \cr & \frac{{dy}}{{dx}} = {D_x}\left[ {\sin {e^{4x}}} \right] \cr & {\text{use the chain rule }}{D_x}\left( {\sin u} \right) = \cos u \cdot {D_x}\left( u \right).{\text{ consider }}u = {e^{4x}} \cr & \frac{{dy}}{{dx}} = \cos {e^{4x}} \cdot {D_x}\left( {{e^{4x}}} \right) \cr & {\text{use the chain rule }}{D_x}\left( {{e^u}} \right) = {e^u} \cdot {D_x}\left( u \right) \cr & \frac{{dy}}{{dx}} = \cos {e^{4x}}\left( {{e^{4x}}} \right){D_x}\left( {4x} \right) \cr & {\text{solve the derivative and simplify}} \cr & \frac{{dy}}{{dx}} = \cos {e^{4x}}\left( {{e^{4x}}} \right)\left( 4 \right) \cr & \frac{{dy}}{{dx}} = 4{e^{4x}}\cos {e^{4x}} \cr} $$
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