BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805
BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Solutions

Chapter 3.4, Problem 18E
To determine

Find the derivative of the given function.

Expert Solution

Answer to Problem 18E

The derivative of the given function is dydt=2e2tcos4t4e2tsin(4t) .

Explanation of Solution

Given:

The given function is y=e2tcos4t .

Calculation:

  y=e2tcos4t

Apply product rule.

  (fg)=f'g+fg'

  dydt=ddt(e2t)cos4t+e2tddt(cos4t)

Apply chain rule.

  df(a)dx=dfdadadx

Let f=ea,a=2t and f=cosa,a=4t

  dydt=ddt(e2t)cos4t+e2tddt(cos4t)dydt=[dda(ea)ddt(2t)]cos4t+e2t[dda(cosa)ddt(4t)]

Use derivative rule.

  ddx(cosx)=sinx and ddx(ex)=ex

  dydt=[ea(2)]cos4t+e2t[sina4]

Substitute the value of a=2t and a=4t .

  dydt=[2e2t]cos4t+e2t[4sin(4t)]dydt=2e2tcos4t4e2tsin(4t)

Hence the derivativeof the given function is dydt=2e2tcos4t4e2tsin(4t) .

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