Find the derivatives of the following at any point of their domains :
`f(x)=(3x^(2)+2)^(2)`

Answer

To find the derivative of the function \( f(x) = (3x^2 + 2)^2 \), we will use the chain rule of differentiation. Here’s a step-by-step solution:

### Step 1: Identify the outer and inner functions
In the function \( f(x) = (3x^2 + 2)^2 \), we can identify:
- The outer function \( g(u) = u^2 \) where \( u = 3x^2 + 2 \)
- The inner function \( h(x) = 3x^2 + 2 \)

### Step 2: Differentiate the outer function
Using the power rule, the derivative of \( g(u) = u^2 \) is:
\[ g'(u) = 2u \]

### Step 3: Differentiate the inner function
Next, we differentiate the inner function \( h(x) = 3x^2 + 2 \):
\[ h'(x) = 6x \]

### Step 4: Apply the chain rule
According to the chain rule:
\[ f'(x) = g'(h(x)) \cdot h'(x) \]
Substituting the derivatives we found:
\[ f'(x) = 2(3x^2 + 2) \cdot (6x) \]

### Step 5: Simplify the expression
Now we simplify the expression:
\[ f'(x) = 12x(3x^2 + 2) \]

### Final Result
Thus, the derivative of the function \( f(x) = (3x^2 + 2)^2 \) is:
\[ f'(x) = 12x(3x^2 + 2) \]