The problem asks to find the derivative of the given power series with respect to $x$, represented as:

$\frac{d}{dx} \left( \sum_{n=0}^\infty a_n x^n \right).$

Solution:

We differentiate term-by-term. Using the power rule, the derivative of $x^n$ with respect to $x$ is $n x^{n-1}$. Therefore, applying this to the series:

$\frac{d}{dx} \left( \sum_{n=0}^\infty a_n x^n \right) = \sum_{n=0}^\infty \frac{d}{dx} \left( a_n x^n \right) = \sum_{n=1}^\infty n a_n x^{n-1}.$

Key Steps:

1. For $n=0$, the term $a_0 x^0$ is constant, so its derivative is zero.
2. The summation starts from $n=1$, as higher-order terms contribute derivatives.

Thus, the resulting power series for the derivative is:

$\sum_{n=1}^\infty n a_n x^{n-1}.$

Would you like a deeper explanation or examples of applications of this operation?

Follow-up Questions:

1. What is the interpretation of differentiating a power series in real-world problems?
2. Can we find the second derivative of the given power series?
3. How does the radius of convergence change after differentiation?
4. What are examples of functions represented by power series and their derivatives?
5. How is this method applied in Taylor or Maclaurin series expansions?

Tip:

When differentiating power series, always check whether the starting index of the series changes (e.g., constant terms drop out).