Question

If $f\left(x\right)=\frac{1}{1-x}$ then the derivative of the composite function $f\left[f\left\{f\right(x\left)\right\}\right]$ is equal to

• A. $0$
• B. $\frac{1}{2}$
• C. $1$
• D. $2$

Answer 1

Answer: (C)

$1$

Clearly, $f\left(x\right)$ is not defined at $x=1$
For $n\in R-\left\{1\right\}$
$g\left(x\right)=f\left[f\right(f\left(x\right)\left)\right]$
$=f\left[f\frac{1}{1-x}\right]=f\left(\frac{1}{1-\frac{1}{1-x}}\right)$
$\Rightarrow g\left(x\right)=f\left(\frac{x-1}{x}\right)=x$
Therefore, $g^{\prime }\left(x\right)=1$ for all $n\in R-\left\{1\right\}$