Find the derivative of the function \(f(x) = \frac{\sqrt{x^3+1}}{x^2+2}\) and simplify your answer by rationalizing the radical expression.

Step 1 ：Step 1: Apply the quotient rule for derivatives, which states that if we have a function in the form of \(f(x) = \frac{g(x)}{h(x)}\), then its derivative is \(f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2}\). In this case, \(g(x) = \sqrt{x^3+1}\) and \(h(x) = x^2+2\).

Step 2 ：Step 2: Find \(g'(x)\) and \(h'(x)\). The derivative of \(g(x)\) is \(\frac{1}{2}(x^3+1)^{-\frac{1}{2}}\cdot 3x^2\), simplifying to \(\frac{3x^2}{2\sqrt{x^3+1}}\). The derivative of \(h(x)\) is \(2x\).

Step 3 ：Step 3: Substitute \(g'(x)\), \(h(x)\), \(g(x)\) and \(h'(x)\) into the quotient rule to obtain \(f'(x) = \frac{(\frac{3x^2}{2\sqrt{x^3+1}})(x^2+2) - (\sqrt{x^3+1})(2x)}{(x^2+2)^2}\).

Step 4 ：Step 4: Simplify this expression to get \(f'(x) = \frac{3x^4 + 6x^2 - 2x\sqrt{x^3+1}}{2(x^2+2)^2\sqrt{x^3+1}}\).

Step 5 ：Step 5: To rationalize the expression, multiply the numerator and the denominator by \(\sqrt{x^3+1}\), we get \(f'(x) = \frac{3x^4\sqrt{x^3+1} + 6x^2\sqrt{x^3+1} - 2x(x^3+1)}{2(x^2+2)^2(x^3+1)}\).

Step 6 ：Step 6: Simplify this expression to get \(f'(x) = \frac{3x^4\sqrt{x^3+1} + 6x^2\sqrt{x^3+1} - 2x^4 - 2x}{2(x^2+2)^2(x^3+1)}\).