d/dx (2x+3)^3/(4x^2-1)^8

f(x) = (2x+3)^3
g(x) = (4x^2 - 1)^8


  = d/dx (2x+3) ^ 3

    f(x) = (2x+3) 
    f'(x) = 2

    g(x) = x ^ 3
    g'(x) = 3x^2

  d/dx g(f(x)) = g'(f(x)) * f'(x)

  = d/dx (2x+3)^3 
  = 3(2x + 3)^2 * 2
  = 6 * (2x + 3) ^2
 

  = d/dx (4x^2 - 1)^8

    f(x) = 4x^2 - 1
    f'(x) = 8x

    g(x) = x ^ 8
    g'(x) = 8x ^ 7

  = d/dx (4x^2 - 1)^8
  = 8(4x^2 - 1)^7 * (8x)

d/dx f(x)/g(x) = (f'g - g'f) / g^2

= ((6(2x + 3)^2)((4x^2 - 1)^8) - (8(4x^2 - 1)^7 * (8x))((2x+3)^3)) / 
  ((4x^2 - 1)^8)^2
= ((6(2x + 3)^2)((4x^2 - 1)^8) - (8(4x^2 - 1)^7 * (8x))((2x+3)^3)) / 
  ((4x^2 - 1)^16)

= (6(2x + 3)^2 * (4x^2 - 1) - 8 * (8x)(2x+3) ^ 3) / ((4x^2 - 1)^9)
= (2(2x+3)^2 * (3(4x^2 - 1) - 32x(2x + 3))) / ...
= (2(2x + 3)^2 * (12x^2 - 3 - 64x^2 - 96x)) / ...
= (2(2x + 3)^2 * -1(52x^2 + 96x + 3) / ...

= (-2(2x + 3)^2 * (52x^2 + 96x + 3) / ((4x^2 - 1)^9)