Find the derivative of the function. \[ g(x)=\frac{3}{x^{6}}+\frac{2}{x^{2}}+9 \sqrt[3]{x} \] \( g^{\prime}(x)=\square \)

To find the derivative \( g'(x) \) of the function \[ g(x) = \frac{3}{x^{6}} + \frac{2}{x^{2}} + 9 \sqrt[3]{x}, \] we'll differentiate each term separately using the power rule. 1. **First Term: \( \frac{3}{x^{6}} \)** \[ \frac{3}{x^{6}} = 3x^{-6} \] The derivative is: \[ \frac{d}{dx}\left(3x^{-6}\right) = 3 \cdot (-6)x^{-7} = -18x^{-7} = -\frac{18}{x^{7}} \] 2. **Second Term: \( \frac{2}{x^{2}} \)** \[ \frac{2}{x^{2}} = 2x^{-2} \] The derivative is: \[ \frac{d}{dx}\left(2x^{-2}\right) = 2 \cdot (-2)x^{-3} = -4x^{-3} = -\frac{4}{x^{3}} \] 3. **Third Term: \( 9 \sqrt[3]{x} \)** \[ 9 \sqrt[3]{x} = 9x^{\frac{1}{3}} \] The derivative is: \[ \frac{d}{dx}\left(9x^{\frac{1}{3}}\right) = 9 \cdot \frac{1}{3}x^{-\frac{2}{3}} = 3x^{-\frac{2}{3}} = \frac{3}{x^{\frac{2}{3}}} \] **Combining all the derivatives:** \[ g'(x) = -\frac{18}{x^{7}} - \frac{4}{x^{3}} + \frac{3}{x^{\frac{2}{3}}} \] **Final Answer:** \[ g^{\prime}(x) = -\frac{18}{\,x^{7}}\, -\, \frac{4}{\,x^{3}}\, +\, \frac{3}{\,x^{\,2/3}\,} \]