Find the derivative offor some constant a.

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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