Calculus: Early Transcendentals (2nd Edition)

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

Chapter 6 - Applications of Integration - 6.10 Hyperbolic Functions - 6.10 Exercises: 29


$$\frac{{dy}}{{dx}} = 6{x^2}\cosh 3x\sinh 3x + 2x{\cosh ^2}3x$$

Work Step by Step

$$\eqalign{ & y = {x^2}{\cosh ^2}3x \cr & {\text{computing }}dy/dx \cr & \frac{{dy}}{{dx}} = \frac{d}{{dx}}\left( {{x^2}{{\cosh }^2}3x} \right) \cr & {\text{by the product rule}} \cr & \frac{{dy}}{{dx}} = {x^2}\frac{d}{{dx}}\left( {{{\cosh }^2}3x} \right) + {\cosh ^2}3x\frac{d}{{dx}}\left( {{x^2}} \right) \cr & {\text{by the chain rule}} \cr & \frac{{dy}}{{dx}} = {x^2}\left( {2\cosh 3x} \right)\frac{d}{{dx}}\left( {\cosh 3x} \right) + {\cosh ^2}3x\frac{d}{{dx}}\left( {{x^2}} \right) \cr & {\text{using basic formulas for differentiation}} \cr & \frac{{dy}}{{dx}} = {x^2}\left( {2\cosh 3x} \right)\left( {\sinh 3x} \right)\left( 3 \right) + {\cosh ^2}3x\left( {2x} \right) \cr & {\text{multiplying}} \cr & \frac{{dy}}{{dx}} = 6{x^2}\cosh 3x\sinh 3x + 2x{\cosh ^2}3x \cr} $$
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