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  4. If f(x)=(xn-an/x-a) for some constant 'a', then f'(a) is

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Q. If $f\left(x\right)=\frac{x^{n}-a^{n}}{x-a}$ for some constant $'a'$, then $f'(a)$ is

Limits and Derivatives

Solution:

We have, $f\left(x\right)=\frac{x^{n}-a^{n}}{x-a}$
$\therefore f'\left(x\right)=\frac{\left(x-a\right) \frac{d}{dx}\left(x^{n}-a^{n}\right)-\left(x^{n}-a^{n}\right) \frac{d}{dx}\left(x-a\right)}{\left(x-a\right)^{2}}$
$=\frac{\left(x-a\right)\left\{nx^{n-1}\right\}-\left(x^{n}-a^{n}\right)\times1}{\left(x-a\right)^{2}}$
$=\frac{n\left(x-a\right)x^{n-1}-\left(x^{n}-a^{n}\right)}{\left(x-a\right)^{2}}$
$\because f'\left(a\right)=\infty$
$\therefore f'\left(a\right)$ does not exist.