To solve for the derivative of the given function $f(x) = x^{\frac{4}{3}}$, we can apply the power rule, which states:

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

Here, $n = \frac{4}{3}$.

Using the power rule:

$f'(x) = \frac{4}{3} \cdot x^{\frac{4}{3} - 1} = \frac{4}{3} \cdot x^{\frac{1}{3}}$

Thus, the derivative is:

$f'(x) = \frac{4}{3} \cdot x^{\frac{1}{3}}$

Would you like more details or have any questions? Here are some related questions:

1. How does the power rule apply to negative exponents?
2. What happens if the exponent is a fraction, like in this case?
3. How do we differentiate functions involving products or quotients?
4. How would you integrate $x^{\frac{4}{3}}$?
5. Can we apply this method to logarithmic or exponential functions?

Tip: Always simplify the exponent first before applying the power rule to ensure accuracy.