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Let's break down the problem step by step.

We have a function \( f(x) \) defined as follows:

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

This represents a polynomial sum where \( p \) runs from 0 to \( n \), and each term is \( x^p \). We are given several tasks:

### Task 4: Calculer \( f'(x) \)
To differentiate the given function with respect to \( x \), we need to differentiate each term of the sum individually. We differentiate \( x^p \) using the power rule:

\[
\frac{d}{dx}(x^p) = p x^{p-1} \quad \text{for} \quad p \geq 1
\]

So, the derivative of \( f(x) \), denoted \( f'(x) \), becomes:

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

Note that the first term \( x^0 = 1 \) has a derivative of 0, which is why the sum starts from \( p = 1 \).

### Task 5: Calculer \( f'(x) \) sans le symbole \( \sum \)
We are asked to express \( f'(x) \) without the summation symbol. To do this, recognize that the original sum represents a geometric series. The function \( f(x) \) can be rewritten as:

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

To differentiate this, apply the quotient rule:

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

Simplifying this will give you the derivative in a non-summation form.

### Task 6: Retrouver l’expression de \( S_n \)
This likely refers to using the previous work to rediscover the sum \( S_n = \sum_{p=0}^{n} x^p \), which can be written as:

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

Let me know if you'd like a more detailed derivation for any of these tasks or if you have specific questions.

### Related Questions:
1. How do we apply the power rule to polynomial terms like \( x^p \)?
2. Can we derive \( f(x) \) in a simpler way using properties of geometric series?
3. What would happen to \( f(x) \) and \( f'(x) \) if \( x = 1 \)?
4. How does the quotient rule apply to more complex functions like \( f(x) \)?
5. Could we generalize this for sums that don't stop at \( n \), like infinite series?

### Tip:
When dealing with sums or series in calculus, always look for ways to express the sum in a simpler, closed form, especially when geometric series are involved.

Calculer f'(x) pour la fonction définie par f(x) = Σ_{p=0}^{n} x^p

Explore how to differentiate the polynomial function f(x) = Σ_{p=0}^{n} x^p step-by-step. Learn about using the power rule and geometric series properties to find f'(x) and express it without summation symbols. This problem is suitable for advanced high school or first-year university students.

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