Question:
Find the derivative of $\frac{x^{n}-a^{n}}{x-a}$ for some constant $a$.
Solution:
Let $f(x)=\frac{x^{n}-a^{n}}{x-a}$
$\Rightarrow f^{\prime}(x)=\frac{d}{d x}\left(\frac{x^{n}-a^{n}}{x-a}\right)$
By quotient rule,
$f^{\prime}(x)=\frac{(x-a) \frac{d}{d x}\left(x^{n}-a^{n}\right)-\left(x^{n}-a^{n}\right) \frac{d}{d x}(x-a)}{(x-a)^{2}}$
$=\frac{(x-a)\left(n x^{n-1}-0\right)-\left(x^{n}-a^{n}\right)}{(x-a)^{2}}$
$=\frac{n x^{n}-a n x^{n-1}-x^{n}+a^{n}}{(x-a)^{2}}$
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