Find the derivative of $x^{n}+a x^{n-1}+a^{2} x^{n-2}+ \dots +a^{n-1} x+a^{n}$ for some fixed real number $a$.

Let $f(x) = x^{n} + a x^{n-1} + a^{2} x^{n-2} + \dots + a^{n-1} x + a^{n}$.
$\frac{d}{dx} f(x) = \frac{d}{dx} (x^{n} + a x^{n-1} + a^{2} x^{n-2} + \dots + a^{n-1} x + a^{n})$.
Using the linearity property of derivatives:
$= \frac{d}{dx}(x^{n}) + a \frac{d}{dx}(x^{n-1}) + a^{2} \frac{d}{dx}(x^{n-2}) + \dots + a^{n-1} \frac{d}{dx}(x) + \frac{d}{dx}(a^{n})$.
Applying the power rule $\frac{d}{dx}(x^{k}) = k x^{k-1}$ and noting that $\frac{d}{dx}(a^{n}) = 0$ since $a$ is a constant:
$= n x^{n-1} + a(n-1) x^{n-2} + a^{2}(n-2) x^{n-3} + \dots + a^{n-1}(1) + 0$.
Thus,$f'(x) = n x^{n-1} + a(n-1) x^{n-2} + a^{2}(n-2) x^{n-3} + \dots + a^{n-1}$.