find the 2nd derivative f(x)=(2x^3 - 3)^4

Answer 1

The function f(x) = (2x³ - 3)⁴

Apply derivative on each side with respect of x.

f ' (x) = 4 (2x³ - 3)³ d/dx [2x³ - 3]

f ' (x) = 4 (2x³ - 3)³ (6x² )

f ' (x) = 24x² (2x³ - 3)³

Again derivative on each side.

Apply product rule in derivatives d/dx (uv) = uv' + vu'.

f ''(x) = 24x²  d/dx (2x³ - 3)³  + (2x³ - 3)³  d/dx (24x²)

f '' (x) = 24x² (3)(2x³ - 3)² (6x² ) + (2x³ - 3)³  (48x)

f '' (x) = 432x⁴(2x³ - 3)² + 48x (2x³ - 3)³

f '' (x) = (2x³ - 3)² [ 432x⁴ + 48x (2x³ - 3)]

f '' (x) = (2x³ - 3)² [ 432x⁴ + 96x⁴ - 144x]

Second derivative of the function is f ''(x) = (2x³ - 3)²  [ 528x⁴ - 144x ]