Question

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

Answer 1

Here f(x) = $x^n+a{x}^{n-1}+a^2{x}^{n-2}+...+a^{n-1}x+a^n\therefore f^{\prime }\left(x\right)=\frac{d}{dx}\left[\begin{array}{c}{x}^n+a{n}^{n-1}+a^2{x}^{n-2}\\ +...+x^{n-1}+a^n\end{array}\right]$

= $\frac{d}{dx}\left(x^n\right)+a\frac{d}{dx}\left(x^{n-1}\right)+a^2\frac{d}{dx}\left(x^{n-2}\right)+...+a^{n-1}\frac{d}{dx}\left(x\right)+\frac{d}{dx}\left(a^n\right)$

= $n{x}^{n-1}+a(n-1)x^{n-2}+a^2(x-2)x^{n-3}+...+a^{n-1}+0$

= $n{x}^{n-1}+a(n-1)x^{n-2}+a^2(n-2)x^{n-3}+...+a^{n-1}$