Find the derivative of the function. $$ g(x)=\frac{3}{x^{4}}+\frac{2}{x^{3}}+6 \sqrt[3]{x} $$ $$ g^{\prime}(x)=\square $$

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To find the derivative of the function $$ g(x) = \frac{3}{x^{4}} + \frac{2}{x^{3}} + 6 \sqrt[3]{x} $$, we'll differentiate each term separately using the power rule.

1. **Rewrite the function with exponents:**
   $$
   g(x) = 3x^{-4} + 2x^{-3} + 6x^{1/3}
   $$

2. **Differentiate each term:**
   - For $$ 3x^{-4} $$:
     $$
     \frac{d}{dx}(3x^{-4}) = 3 \cdot (-4)x^{-5} = -12x^{-5}
     $$
   - For $$ 2x^{-3} $$:
     $$
     \frac{d}{dx}(2x^{-3}) = 2 \cdot (-3)x^{-4} = -6x^{-4}
     $$
   - For $$ 6x^{1/3} $$:
     $$
     \frac{d}{dx}(6x^{1/3}) = 6 \cdot \frac{1}{3}x^{-2/3} = 2x^{-2/3}
     $$

3. **Combine the derivatives:**
   $$
   g'(x) = -12x^{-5} - 6x^{-4} + 2x^{-2/3}
   $$

4. **Optional (rewrite with positive exponents):**
   $$
   g'(x) = -\frac{12}{x^{5}} - \frac{6}{x^{4}} + \frac{2}{x^{2/3}}
   $$

**Final Answer:**
$$
g^{\prime}(x) = -12\,x^{-5}\ -\ 6\,x^{-4}\ +\ 2\,x^{-2/3}
$$
$$ g^{\prime}(x) = -12x^{-5} - 6x^{-4} + 2x^{-2/3} $$

Find the derivative of the function. \[ g(x)=\frac{3}{x^{4}}+\frac{2}{x^{3}}+6 \sqrt[3]{x} \] \( g^{\prime}(x)=\square \)

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