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 33E
To determine

Find the derivative of the given function.

Expert Solution

Answer to Problem 33E

The derivative of the given function is dydx=2cot(sinx)csc2(sinx)cosx .

Explanation of Solution

Given:

The given function is y=cot2(sinθ) .

Calculation:

  y=cot2(sinθ)

Apply chain rule.

  df(a)dx=dfdadadx

Let f=a2,a=cot(sinx)

  dydx=dda(a2)ddx(cot(sinx))

Use derivative rule.

  ddx(xn)=nxn1 .

  dydx=2addx(cot(sinx))

Substitute the value of a=cot(sinx) .

  dydx=2cot(sinx)ddx(cot(sinx))

Apply chain rule.

  df(a)dx=dfdadadx

Let f=cot(a),a=sinx

  dydx=2cot(sinx)dda(cota)ddx(sinx)

Use derivative rule.

  ddx(cotx)=csc2xandddx(sinx)=cosx .

  dydx=2cot(sinx)csc2acosx

Substitute the value of a=sinx .

  dydx=2cot(sinx)csc2(sinx)cosxdydx=2cot(sinx)csc2(sinx)cosx

Hence the derivativeof the given function is dydx=2cot(sinx)csc2(sinx)cosx .

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