Question

If $g$ is the inverse of a function $f$ and$f\text{'}\left(x\right)=\frac{1}{1+x^5}$, then $g\text{'}\left(x\right)$ is equal to

• A. $1+x^5$
• B. $5x^4$
• C. $\frac{1}{1+{\left(g\left(x\right)\right)}^5}$
• D. $1+{\left(g\left(x\right)\right)}^5$

Answer 1

The correct option is C

$\frac{1}{1+{\left(g\left(x\right)\right)}^5}$

Explanation for the correct option:

Finding the value of $g\text{'}\left(x\right)$:

Since, $g$ is the inverse of a function $f$

So,

$fog\left(x\right)=x$

Differentiate the above function with respect to $x$.

$\begin{array}{rcl}f\text{'}\left(g\left(x\right)\right)·g\text{'}\left(x\right)& =& 1\left[\because \frac{d\left(fog\right)\left(x\right)}{dx}=f\text{'}\left(g\left(x\right)\right)·g\text{'}\left(x\right)\right]\\ & \Rightarrow & g\text{'}\left(x\right)=\frac{1}{f\text{'}\left(g\left(x\right)\right)}\\ & =& \frac{1}{1+{\left(g\left(x\right)\right)}^5}\end{array}$

Hence, the correct option is C.