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!

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!