Find \( f^{\prime}(x), f^{\prime \prime}(x) \), and \( f^{\prime \prime \prime}(x) \). \( f(x)=\frac{1}{x^{3}} \) \( f^{\prime}(x)=\square \)

To find the derivatives of the function \( f(x) = \frac{1}{x^{3}} \), we start with the first derivative: 1. **First Derivative**: Using the power rule, we can rewrite the function as \( f(x) = x^{-3} \). Now applying the derivative: \[ f^{\prime}(x) = -3x^{-4} = -\frac{3}{x^{4}} \] So, \( f^{\prime}(x) = -\frac{3}{x^{4}} \). 2. **Second Derivative**: Next, we take the derivative of the first derivative: \[ f^{\prime \prime}(x) = -3 \cdot (-4)x^{-5} = 12x^{-5} = \frac{12}{x^{5}} \] So, \( f^{\prime \prime}(x) = \frac{12}{x^{5}} \). 3. **Third Derivative**: Finally, we take the derivative of the second derivative: \[ f^{\prime \prime \prime}(x) = 12 \cdot (-5)x^{-6} = -60x^{-6} = -\frac{60}{x^{6}} \] So, \( f^{\prime \prime \prime}(x) = -\frac{60}{x^{6}} \). In summary: - \( f^{\prime}(x) = -\frac{3}{x^{4}} \) - \( f^{\prime \prime}(x) = \frac{12}{x^{5}} \) - \( f^{\prime \prime \prime}(x) = -\frac{60}{x^{6}} \)