Deermine the derivative of f(x)=[(3x^2+1)^3+1]^2. Express your answer in simplified form.

For this I got 54x^6+162x^5+54x^4+108x^3+18x^2+18x+2

The chain rule gives

2[(3x^2+1)^3+1] * 3(3x^2+1)^2 * (6x)

This is an 11th-degree polynomial, so I suspect your answer is wrong.

To determine the derivative of the given function f(x)=[(3x^2+1)^3+1]^2, we can use the chain rule and the power rule for differentiation. Here are the step-by-step instructions on how to get the answer:

Step 1: Start by taking the derivative of the outer function:
Let's call the outer function u(x) = [(3x^2+1)^3+1]^2.
To take the derivative of the outer function, we apply the power rule:
u'(x) = 2[(3x^2+1)^3+1]^1 * (d/dx)[(3x^2+1)^3+1].

Step 2: Now, let's focus on finding the derivative of the inner function (3x^2+1)^3+1.
To take the derivative of this inner function, we will apply the chain rule.
Let's call the inner function v(x) = (3x^2+1)^3+1.
To find the derivative of the inner function, we apply the power rule and the chain rule:
(d/dx)[(3x^2+1)^3+1] = (d/dx)(v(x)) = (d/dx)(v(x))^3 * (d/dx)[(3x^2+1)].

Step 3: To determine the derivative of (d/dx)[(3x^2+1)], we use the power rule for differentiation:
(d/dx)[(3x^2+1)] = 6x.

Step 4: Substitute the derivative of (d/dx)[(3x^2+1)] back into the chain rule for (d/dx)[(3x^2+1)^3+1]:
(d/dx)[(3x^2+1)^3+1] = 3(3x^2+1)^2 * 6x.

Step 5: Substitute the derivative of (d/dx)[(3x^2+1)^3+1] back into the derivative of the outer function u'(x):
u'(x) = 2[(3x^2+1)^3+1] * 3(3x^2+1)^2 * 6x.

Step 6: Simplify the expression:
u'(x) = 36x(3x^2+2)^2[(3x^2+1)^3+1].

Therefore, the derivative of f(x)=[(3x^2+1)^3+1]^2 is u'(x) = 36x(3x^2+2)^2[(3x^2+1)^3+1].