Question

Find the derivative of $\frac{x^n-a^n}{x-a}$ for some constant $a$

Answer 1

Let $f\left(x\right)=\frac{x^n-a^n}{x-a}$
$\Rightarrow f^{{}^{\prime }}\left(x\right)=\frac{d}{dx}\left(\frac{x^n-a^n}{x-a}\right)$
By quotient rule
$f^{{}^{\prime }}\left(x\right)=\frac{(x-a)\frac{d}{dx}(x^n-a^n)-(x^n-a^n)\frac{d}{dx}(x-a)}{{(x-a)}^2}$
= $\frac{(x-a)(n{x}^{n-1}-0)-(x^n-a^n)}{{(x-a)}^2}$
= $\frac{n{x}^n-an{x}^{n-1}-x^n+a^n}{{(x-a)}^2}$