If `f(x)=3sqrt((1+x^(2))^(4))`, find f'(1).

Answer

To find \( f'(1) \) for the function \( f(x) = 3\sqrt{(1+x^2)^4} \), we will follow these steps:

### Step 1: Simplify the Function
First, we simplify the function:
\[ f(x) = 3\sqrt{(1+x^2)^4} \]
Since \( \sqrt{a^4} = a^2 \), we can rewrite the function as:
\[ f(x) = 3(1+x^2)^2 \]

### Step 2: Differentiate the Function
Next, we differentiate \( f(x) \):
\[ f'(x) = \frac{d}{dx}[3(1+x^2)^2] \]
Using the chain rule, we have:
\[ f'(x) = 3 \cdot 2(1+x^2) \cdot \frac{d}{dx}(1+x^2) \]
Now, differentiate \( 1+x^2 \):
\[ \frac{d}{dx}(1+x^2) = 2x \]
Substituting this back, we get:
\[ f'(x) = 3 \cdot 2(1+x^2) \cdot 2x = 12x(1+x^2) \]

### Step 3: Evaluate the Derivative at \( x = 1 \)
Now we need to find \( f'(1) \):
\[ f'(1) = 12 \cdot 1 \cdot (1 + 1^2) = 12 \cdot 1 \cdot (1 + 1) = 12 \cdot 1 \cdot 2 = 24 \]

### Final Answer
Thus, the value of \( f'(1) \) is:
\[ \boxed{24} \]