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 37E
To determine

Find the first and second derivative of the given function.

Expert Solution

Answer to Problem 37E

The first derivative is y'=2xsin(x2) and the second derivative is y"=2[2x2cos(x2)+sin(x2)] .

Explanation of Solution

Given:

The given function is y=cos(x2) .

Calculation:

  y=cos(x2)

Apply chain rule.

  df(a)dx=dfdadadx

Let f=cosa,a=x2

  dydx=dda(cosa)ddx(x2)

Use derivative rule.

  ddx(cosa)=sina and ddx(xn)=nxn1

  dydx=sina2x

Substitute the value of a=x2 .

  y'=2xsin(x2)

Use product rule.

  (fg)'=fg'+gf'

  y"=2[xddx{sin(x2)}+sin(x2)ddx(x)]

Apply chain rule.

  df(a)dx=dfdadadx

Let f=sina,a=x2

  y"=2[xdda(sina)ddx(x2)+sin(x2)ddx(x)]

Use derivative rule.

  ddx(sina)=cosa and ddx(xn)=nxn1

  y"=2[xcosa2x+sin(x2)]

Substitute the value of a=x2 .

  y"=2[2x2cos(x2)+sin(x2)]

Hence the first derivativeis y'=2xsin(x2) and the second derivative is y"=2[2x2cos(x2)+sin(x2)] .

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