https://file.aiquickdraw.com/mathbot/file/1727030878686tmxb1te1.jpg

The image shows a function \( g(x) = \sqrt[3]{x} \) followed by some steps of simplification and differentiation.

### Step-by-step explanation:

1. **Function definition:**
   \[
   g(x) = \sqrt[3]{x}
   \]
   This is the cube root of \(x\), which can also be written as:
   \[
   g(x) = x^{1/3}
   \]

2. **Differentiation:**
   To find \( g'(x) \), we use the power rule:
   \[
   \frac{d}{dx}(x^n) = n \cdot x^{n-1}
   \]
   Applying this to \( g(x) = x^{1/3} \):
   \[
   g'(x) = \frac{1}{3} x^{-\frac{2}{3}}
   \]

   This simplifies to:
   \[
   g'(x) = \frac{1}{3 \sqrt[3]{x^2}}
   \]

Let me know if you'd like more details or further clarification!

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Here are 5 related questions to deepen your understanding:

1. How would you differentiate other radical functions like \( \sqrt{x} \) or \( \sqrt[4]{x} \)?
2. What is the significance of the power rule in calculus?
3. Can you explain why the derivative of \( x^n \) changes when \( n \) is a fraction?
4. How would the derivative change if we were dealing with a higher root, like \( \sqrt[5]{x} \)?
5. How do you interpret the derivative \( g'(x) \) in terms of rates of change?

**Tip:** When differentiating functions involving roots, always try to express the roots as fractional exponents for easier application of rules like the power rule!

Given the function g(x) = ∛x, find its derivative.

Learn how to differentiate the cube root function g(x) = ∛x. This step-by-step guide explains the use of the power rule for differentiation and provides a detailed solution suitable for high school calculus students.

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