Question:
Find the derivative of $x^{n}+a x^{n-1}+a^{2} x^{n-2}+\ldots+a^{n-1} x+a^{n}$ for some fixed real number $a$.
Solution:
Let $f(x)=x^{n}+a x^{n-1}+a^{2} x^{n-2}+\ldots+a^{n-1} x+a^{n}$
$\therefore f^{\prime}(x)=\frac{d}{d x}\left(x^{n}+a x^{n-1}+a^{2} x^{n-2}+\ldots+a^{n-1} x+a^{n}\right)$
$=\frac{d}{d x}\left(x^{n}\right)+a \frac{d}{d x}\left(x^{n-1}\right)+a^{2} \frac{d}{d x}\left(x^{n-2}\right)+\ldots+a^{n-1} \frac{d}{d x}(x)+a^{n} \frac{d}{d x}(1)$
On using theorem $\frac{d}{d x} x^{n}=n x^{n-1}$, we obtain
$f^{\prime}(x)=n x^{n-1}+a(n-1) x^{n-2}+a^{2}(n-2) x^{n-3}+\ldots+a^{n-1}+a^{n}(0)$
$=n x^{n-1}+a(n-1) x^{n-2}+a^{2}(n-2) x^{n-3}+\ldots+a^{n-1}$