Find the indicated derivative for the function.
\[
f^{\prime \prime}(x) \text { for } f(x)=3 x^{5}-9 x^{2}+2 x-3
\]
Therefore, the second derivative of the function \(f(x) = 3x^{5} - 9x^{2} + 2x - 3\) is \(\boxed{60x^{3} - 18}\).
Step 1 :The given function is \(f(x) = 3x^{5} - 9x^{2} + 2x - 3\).
Step 2 :First, find the first derivative of the function, \(f'(x)\), by applying the power rule for differentiation, which states that the derivative of \(x^n\) is \(nx^{n-1}\).
Step 3 :So, \(f'(x) = 15x^{4} - 18x + 2\).
Step 4 :Then, differentiate \(f'(x)\) again to find the second derivative, \(f''(x)\).
Step 5 :So, \(f''(x) = 60x^{3} - 18\).
Step 6 :Therefore, the second derivative of the function \(f(x) = 3x^{5} - 9x^{2} + 2x - 3\) is \(\boxed{60x^{3} - 18}\).