Find the second derivative of the function.
Select the correct answer.
$$f ( x ) = x \left( 3 x ^ { 2 } - 1 \right) ^ { 4 }$$
A) $4 x \left( 3 x ^ { 2 } - 1 \right) ^ { 3 }$
B) $72 x \left( 3 x ^ { 2 } - 1 \right) ^ { 2 } \left( 9 x ^ { 2 } - 1 \right)$
C) $12 x \left( 3 x ^ { 2 } - 1 \right) ^ { 2 }$
D) $\left( 27 x ^ { 2 } - 1 \right) \left( 3 x ^ { 2 } - 1 \right) ^ { 3 }$

Question

Quizplus · Accepted Answer

The first derivative of the given function is found by applying the product rule and the chain rule:
$$f'(x)=3x^2(3x^2-1)^3+4x^2(3x^2-1)^3$$
$$=x(3x^2-1)^3(7\cdot 3x^2-1)$$
To find the second derivative, we can use the product rule again, but it will be easier to simplify the expression for the first derivative first:
$$f'(x)=x(3x^2-1)^3(7\cdot 3x^2-1)=21x^3(3x^2-1)^3$$
Applying the product rule gives:
$$f''(x)=63x^2(3x^2-1)^3+6x\cdot 3(3x^2-1)^2\cdot 42x^2$$
$$=3 x \left(3x^2-1\right)^2 \left(72x^2-7\right)$$
which is equivalent to answer choice B.

Find the Second Derivative of the Function f(x)=x(3x2−1)4f ( x ) = x \left( 3 x ^ { 2 } - 1 \right) ^ { 4 }f(x)=x(3x2−1)4

Find the Second Derivative of the Function $f ( x ) = x \left( 3 x ^ { 2 } - 1 \right) ^ { 4 }$