Find the derivative of the Function: P(x) = (x - 3cosx)(x + 3cosx)

Question

I know you have to use the product rule(uv) which is u times the derivative of v plus v times the derivative of v. But how do you find the derivative of (x + 3cosx) and the derivative of (x - 3cosx)?

Answer 1

P(x) = (x - 3cosx)(x + 3cosx) = x^2 - 9cos^2(x)

dp/dx = 2x - 9d(cos^2x) = 2x - 9*2cosx*d(cosx) =

                                     = 2x - 9*2cosx(- sinx) =

                                        2x +  9*2sinxcosx =

                                        2x + 9sin(2x)

you can use the formula

(a - b)(a + b) = a^2 - b^2

2sinxcosx = sin(2x)

or you can use the  product rule

 dp/dx = (d( x - 3cosx))(x + 3cosx) + ( x - 3cosx)(d(x + 3cosx)) =

                 (1 + 3sinx)(x + 3cosx) + ( x - 3cosx)(1 - 3sinx) =

   x + 3cosx + 3xsinx + 9 sinxcosx + x -3cosx - 3xsinx + 9 sinxcosx =

  2x + 18 sinxcosx = 2x + 9*2sinxcosx=  2x + 9 sin(2x)