Use logarithmic differentiation to find the derivative of the function. y = (x3 + 2)2(x5 + 4)4

Ln y = ln (x^3 + 2)^2 (x^5 + 4)^4

ln y  = 2 ln (x^3 + 2)  + 4 ln(x^5 + 4)

y' 1/y =  2 * 3x^2 * 1 / ( x^3 + 2) + 4 * 5x^4 * 1 / (x^5 + 4)

         =   6x^2 / (x^3 + 2) + 20x^4 / ( x^5 + 4)

   y'     =    y * (6x^2 / (x^3 + 2) + 20x^4 / ( x^5 + 4))

          =   6x^2 (x^3 + 2)^2 (x^5 + 4)^4 + 20x^4 (x^3 + 2)^2 (x^5 + 4)^4
               
                                 (x^3 + 2) (x^5 + 4)

          =  6x^2 ( x^3 + 2)(x^5 + 4)^3 + 20x^4(x^3 + 2)(x^5 + 4)^3

          =  2x^2(3 + 10x^2)(x^3 + 2)(x^4 + 5)^3  Answer