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 38E
To determine

Find the first and second derivative of the given function.

Expert Solution

Answer to Problem 38E

The first derivative is y'=sin2x and the second derivative is y"=2cos2x .

Explanation of Solution

Given:

The given function is y=cos2x .

Calculation:

  y=cos2x

Apply chain rule.

  df(a)dx=dfdadadx

Let f=a2,a=cosx

  dydx=dda(a2)ddx(cosx)

Use derivative rule.

  ddx(cosx)=sinx and ddx(xn)=nxn1

  dydx=2asinx

Substitute the value of a=cosx .

  y'=2cosxsinxy'=sin2x

Use product rule.

  (fg)'=fg'+gf'

  y"=2[cosxddxsinx+sinxddxcosx]

Use derivative rule.

  ddx(sinx)=cosx and ddx(cosx)=sinx

  y"=2[cosxcosx+sinx(sinx)]y"=2[cos2xsin2x]y"=2cos2x

Hence the first derivativeis y'=sin2x and the second derivative is y"=2cos2x .

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