can you simplify the answer to this explanation? "Answer: Answer: The derivative of the function \( f(x) = \frac{3x}{(x^2 + 2)^2} \) is \( f'(x) = \frac{3(x^2 + 2)^2 - 12x^2(x^2 + 2)}{(x^2 + 2)^4} \). Explanation: To differentiate the function \( f(x) = \frac{3x}{(x^2 + 2)^2} \), we will use the quotient rule. The quotient rule states that if you have a function \( \frac{u}{v} \), its derivative is given by: \[ \left(\frac{u}{v} ight)' = \frac{u'v - uv'}{v^2} \] where \( u = 3x \) and \( v = (x^2 + 2)^2 \). 1. **Differentiate \( u = 3x \)**: - The derivative \( u' = 3 \). 2. **Differentiate \( v = (x^2 + 2)^2 \)**: - Use the chain rule. Let \( w = x^2 + 2 \), so \( v = w^2 \). - The derivative of \( w^2 \) is \( 2w \cdot w' \). - The derivative of \( w = x^2 + 2 \) is \( w' = 2x \). - Therefore, \( v' = 2(x^2 + 2) \cdot 2x = 4x(x^2 + 2) \). 3. **Apply the Quotient Rule**: \[ f'(x) = \frac{3(x^2 + 2)^2 - 3x \cdot 4x(x^2 + 2)}{((x^2 + 2)^2)^2} \] 4. **Simplify**: - The numerator becomes \( 3(x^2 + 2)^2 - 12x^2(x^2 + 2) \). - The denominator is \( (x^2 + 2)^4 \). Thus, the derivative is: \[ f'(x) = \frac{3(x^2 + 2)^2 - 12x^2(x^2 + 2)}{(x^2 + 2)^4} \]