Find the derivative of the Function: P(x)=(x−3cos⁡x)(x+3cos⁡x)

Question

Find the derivative of the Function:   P(x)=(x−3cos⁡x)(x+3cos⁡x)

dessinemoie · Accepted Answer

P(x)=(x−3cos⁡x)(x+3cos⁡x)=x2−9cos2(x)  dpdx=2x−9d(cos2x)=2x−9⋅2cos⁡x⋅d(cos⁡x)=  =2x−9⋅2cos⁡x(−sin⁡x)  =2x+9⋅2sin⁡xcos⁡x  =2x+9sin⁡(2x)  you can use the formula  (a−b)(a+b)=a2−b2  2sin⁡xcos⁡x=sin⁡(2x)  or you can use the  product rule  dpdx=(d(x−3cos⁡x))(x+3cos⁡x)+(x−3cos⁡x)(d(x+3cos⁡x))=  (1+3sin⁡x)(x+3cos⁡x)+(x−3cos⁡x)(1−3sin⁡x)=  x+3cos⁡x+3xsin⁡x+9sin⁡xcos⁡x+x−3cos⁡x−3xsin⁡x+9sin⁡xcos⁡x=  2x+18sin⁡xcos⁡x=2x+9⋅2sin⁡xcos⁡x=2x+9sin⁡(2x)

Find the derivative of the Function: P(x)=(x−3cos⁡x)(x+3cos⁡x)

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