Question

If $\phi \left(x\right)$ is the inverse of the function $f\left(x\right)$ and $f\text{'}\left(x\right)=\frac{1}{\left(1+x^5\right)}$, then $\frac{d}{dx}\phi \left(x\right)$ is

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

Answer 1

The correct option is C

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

Explanation for the correct option

Since the function $\phi \left(x\right)$ is equal to the inverse function of $f\left(x\right)$,

$\begin{array}{rcl}\therefore \phi \left(x\right)& =& f^{-1}\left(x\right)\\ & \Rightarrow & f\left(\phi \left(x\right)\right)=x\end{array}$

Now, differentiate both the sides with respect to $x$, we get

$f\text{'}\left(\phi \left(x\right)\right)·\phi \text{'}\left(x\right)=1$

From this, we have

$\phi \text{'}\left(x\right)=\frac{1}{f\text{'}\left(\phi \left(x\right)\right)}\dots \left(1\right)$

Now, it is given that $f\text{'}\left(x\right)=\frac{1}{\left(1+x^5\right)}$ then,

$f\text{'}\left(\phi \left(x\right)\right)=\frac{1}{\left(1+{\left\{\phi \left(x\right)\right\}}^5\right)}$

Replace this in $\left(1\right)$,

$\begin{array}{rcl}\phi \text{'}\left(x\right)& =& \frac{1}{f\text{'}\left(\phi \left(x\right)\right)}\\ & =& \frac{1}{\frac{1}{\left(1+{\left\{\phi \left(x\right)\right\}}^5\right)}}\\ & =& \left(1+{\left\{\phi \left(x\right)\right\}}^5\right)\end{array}$

Hence, the correct option is (C).