Find the derivative of `x^(n)+ax^(n-1)+a^(2)x^(n-2)+..........+a^(n-1)x+a^(n)` for some fixed real number a.

Explanation

To find the derivative of the expression \( f(x) = x^n + ax^{n-1} + a^2x^{n-2} + \ldots + a^{n-1}x + a^n \), we can follow these steps:

### Step 1: Identify the terms in the expression
The expression consists of \( n+1 \) terms, where each term follows the pattern \( a^k x^{n-k} \) for \( k = 0, 1, 2, \ldots, n \).

### Step 2: Write the expression in summation form
We can express the function \( f(x) \) in summation notation:
\[
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