Question

The $n$th derivative of the function $f\left(x\right)=\frac{1}{1-x^2}[$where$xϵ(-1,1)]$ at the point $x=0$ where $n$ is even is

• A. 0
• B. $n!$
• C. $n^n{C}_2$
• D. $2^n{C}_2$

Answer 1

The correct option is B $n!$

$f\left(x\right)=\frac{1}{1-x^2}=1+x^2+x^4+x^6+....x^n+\cdots \infty$, where $|x|<1$
$\Rightarrow f^n\left(x\right)=n!+(n+1)!x+.......\infty$
$\therefore f^n\left(0\right)=n!$, where $n$ is even