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Let's analyze the mathematical problem step by step.

The given function \( f(x) \) is defined as:

\[
f(x) = \sum_{p=0}^{n} x^p
\]

Where \( x \in \mathbb{R} \) and \( f(x) \) is a polynomial, so it is continuous and differentiable on \( \mathbb{R} \).

The problem consists of several parts:

### 4. Calculate \( f'(x) \) for real \( x \).

Since \( f(x) = \sum_{p=0}^{n} x^p \), differentiating this with respect to \( x \):

\[
f'(x) = \sum_{p=0}^{n} p x^{p-1}
\]

This is the general expression for the derivative of the function.

### 5. Calculate \( f'(x) \) in a different way for \( x \neq 1 \) after writing \( f(x) \) without the summation symbol.

The function \( f(x) = \sum_{p=0}^{n} x^p \) can be written as a closed-form expression for a geometric series:

\[
f(x) = \frac{x^{n+1} - 1}{x - 1} \quad \text{for} \quad x \neq 1
\]

Now, differentiate this expression:

\[
f'(x) = \frac{(x - 1)(n+1)x^n - (x^{n+1} - 1)}{(x - 1)^2}
\]

Simplify this expression to find \( f'(x) \) in this case.

### 6. Use the previous result to retrieve the expression for \( S_n \).

The problem asks to find the expression for \( S_n \), which might refer to the sum of powers of \( x \) from \( 0 \) to \( n \), i.e., the geometric series sum:

\[
S_n = \sum_{p=0}^{n} x^p = \frac{x^{n+1} - 1}{x - 1} \quad \text{for} \quad x \neq 1
\]

Thus, the expression for \( S_n \) is the same closed-form expression we derived in part 5.

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Would you like more details or clarification on any part?

Here are five questions to consider that expand upon this:

1. How would the derivative change if \( f(x) \) had a different upper bound for the summation, say \( \sum_{p=1}^{n} x^p \)?
2. Can you explain why the closed-form expression for a geometric series works, especially for \( x \neq 1 \)?
3. How would you handle differentiating \( f(x) \) if the power of \( x \) were negative in some terms of the series?
4. What happens to the sum \( S_n \) when \( x = 1 \), and how can you deal with the indeterminate form?
5. How does this problem connect to other areas of calculus, such as finding the area under a curve?

**Tip**: When dealing with geometric series, always check for conditions like \( x \neq 1 \) to avoid undefined behavior or indeterminate forms.

Soit f la fonction définie sur ℝ par f(x) = ∑(x^p) de p=0 à n. Calculer la dérivée f'(x), puis f'(x) pour x ≠ 1 sans le symbole de somme. Retrouver l'expression de S_n.

Explore how to differentiate a polynomial summation function f(x) = ∑_{p=0}^{n} x^p and retrieve its geometric series representation for x ≠ 1. Step-by-step guide suitable for undergraduate calculus students.

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