Question

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

• A. $1$
• B. $0$
• C. Does not exist
• D. $\frac{1}{2}$

Answer 1

Answer: (C)

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