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Q.
For some constant $a$, the derivative of $\frac{x^n-a^n}{x-a}$ is
Limits and Derivatives
Solution:
Let $y=\frac{x^n-a^n}{x-a}$
Differentiating $y$ w.r.t $x$, we get
$\frac{d y}{d x}=\frac{(x-a) \frac{d}{d x}\left(x^n-a^n\right)-\left(x^n-a^n\right) \frac{d}{d x}(x-a)}{(x-a)^2}$
$=\frac{(x-a)\left[n x^{n-1}-0\right]-\left(x^n-a^n\right)(1-0)}{(x-a)^2}$
$=\frac{(x-a) n x^{n-1}-x^n+a^n}{(x-a)^2}$
$=\frac{x \times n x^{n-1}-a n x^{n-1}-x^n+a^n}{(x-a)^2}$
$=\frac{n x^n-a n x^{n-1}-x^n+a^n}{(x-a)^2}$