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, Problem 4P
To determine

To show: The derivative of ddx(sin2x1+cotx+cos2x1+tanx)=cos2x .

Expert Solution

Explanation of Solution

Proof:

Consider f(x)=sin2x1+cotx+cos2x1+tanx.

f(x)=sin2x1+cosxsinx+cos2x1+sinxcosx=sin2xsinx+cosxsinx+cos2xcosx+sinxcosx=sin3xsinx+cosx+cos3xcosx+sinx=sin3x+cos3xsinx+cosx

Use the formula a3+b3=(a+b)(a2ab+b3) and simplify further,

f(x)=(sinx+cosx)(sin2xsinxcosx+cos2x)sinx+cosx=1sinxcosx                                         (Qsin2x+cos2x=1)=112(2sinxcosx)=112sin2x                                           (Q2sinxcosx=sin2x)

Obtain the derivative of f(x).

ddx(sin2x1+cotx+cos2x1+tanx)=ddx(112sin2x)=ddx(1)12ddx(sin2x)=012cos2x2=cos2x

Hence, ddx(sin2x1+cotx+cos2x1+tanx)=cos2x proved.

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