Question

For any constant real number $a$, find the derivative of:
$x^n+a{x}^{n-1}+a^2{x}^{n-2}+...+a^{n-1}x+a^n$.

Answer 1

Let $f\left(x\right)=x^n+a{x}^{n-1}+a^2{x}^{n-2}+\cdot \cdot \cdot +a^{n-1}x+a^n$
$\therefore f^{{}^{\prime }}\left(x\right)=\frac{d}{dx}(x^n+a{x}^{n-1}+a^2{x}^{n-2}+\cdot \cdot \cdot +a^{n-1}x+a^n)$
= $\frac{d}{dx}\left(x^n\right)+a\frac{d}{dx}\left(x^{n-1}\right)+a^2\frac{d}{dx}\left(x^{n-2}\right)+\cdot \cdot \cdot +a^{n-1}\frac{d}{dx}\left(x\right)+a^n\frac{d}{dx}\left(1\right)$
$=n{x}^{n-1}+a(n-1)x^{n-2}+a^2(n-2)x^{n-3}+\cdot \cdot \cdot +a^{n-1}+a^n\left(0\right)$
$\Rightarrow f^{\prime }\left(x\right){\mid }_{x=a}=n{x}^{n-1}+a(n-1)x^{n-2}+a^2(n-2)x^{n-3}+\cdot \cdot \cdot +a^{n-1}$