The function you want to differentiate is the cube root of $x$, which can be expressed as:

$f(x) = \sqrt[3]{x} = x^{1/3}$

To find the derivative of this function, we use the power rule for derivatives. The power rule states that for any function $x^n$, its derivative is given by:

$\frac{d}{dx}(x^n) = n x^{n-1}$

Applying this rule to $f(x) = x^{1/3}$:

$f'(x) = \frac{1}{3} x^{1/3 - 1} = \frac{1}{3} x^{-2/3}$

Thus, the derivative of $f(x) = \sqrt[3]{x}$ is:

$f'(x) = \frac{1}{3} x^{-2/3} = \frac{1}{3 \sqrt[3]{x^2}}$

Would you like more details or have any other questions?

Here are 5 related questions you can explore:

1. What is the second derivative of $f(x) = \sqrt[3]{x}$?
2. How do you compute the derivative of $f(x) = \sqrt[3]{x^2}$?
3. How do you differentiate $f(x) = x^{m/n}$ for any general rational power?
4. Can you apply the chain rule to differentiate $\sqrt[3]{g(x)}$?
5. How do you compute the derivative of implicit functions involving cube roots?

Tip: When dealing with fractional powers, always check if applying logarithmic differentiation simplifies the process.