To find the series expansion for the derivative of the given function $f(x)$, we will differentiate the given series expansion of $f(x)$ term-by-term. The given series expansion is:

$f(x) = \sum_{n=0}^{\infty} x^{5n}$

We need to differentiate this series term-by-term to find the series expansion for $f'(x)$:

$f'(x) = \frac{d}{dx} \left( \sum_{n=0}^{\infty} x^{5n} \right)$

Using the linearity of differentiation and the fact that we can interchange the summation and differentiation (since the series is uniformly convergent), we get:

$f'(x) = \sum_{n=0}^{\infty} \frac{d}{dx} \left( x^{5n} \right)$

To differentiate each term, we use the power rule of differentiation:

$\frac{d}{dx} \left( x^{5n} \right) = 5n x^{5n-1}$

Therefore, we have:

$f'(x) = \sum_{n=0}^{\infty} 5n x^{5n-1}$

Since the term for $n = 0$ is zero (because $5 \cdot 0 \cdot x^{-1} = 0$), we can start the summation from $n = 1$:

$f'(x) = \sum_{n=1}^{\infty} 5n x^{5n-1}$

Thus, the series expansion for the derivative of $f(x)$ is:

$\boxed{f'(x) = \sum_{n=1}^{\infty} 5n x^{5n-1}}$

Do you want details or have any questions?

Here are 8 related questions that expand on the previous information:

1. How can we verify the convergence of the differentiated series?
2. What is the radius of convergence for the series expansion of $f'(x)$?
3. How would the differentiation change if the series started at $n = 1$ instead of $n = 0$?
4. Can the method of term-by-term differentiation be applied to non-uniformly convergent series?
5. What are the conditions under which term-by-term differentiation is valid?
6. How does the derivative of the series expansion compare with the original function $f(x)$?
7. What is the significance of the power series in this context?
8. How can we express the derivative $f'(x)$ in a closed form if possible?

Tip: When differentiating a power series term-by-term, ensure that the series converges uniformly on the interval where you differentiate. This guarantees the validity of interchanging the summation and differentiation operations.