Question

If $g$ is the inverse of the function $f$ and $f'\left( x \right) = \dfrac{1}{{\left( {1 + {x^5}} \right)}},$ then $g'\left( x \right)$ is equal to
$\left( 1 \right)1 + {x^5}$
$\left( 2 \right)5{x^4}$
\left( 3 \right)\dfrac{1}{{\left( {1 + {{\left\\{ {g\left( x \right)} \right\\}}^5}} \right)}}
\left( 4 \right)1 + {\left\\{ {g\left( x \right)} \right\\}^5}

Answer 1

Here in the question we have two functions $f\left( x \right){\text{ and g}}\left( x \right)$ and it is given that the function $g\left( x \right)$ is the inverse of the function $f\left( x \right)$, so we should have some knowledge about inverse functions. Inverse functions are generally functions that “reverse” each other. For example: If one function takes input as $m$ and gives output as $n$ , then the other function should take input as $n$ and must give the output as $m$ , if this property satisfies then the functions are called inverse functions of each other.

Complete step by step answer:
Here $g$ is the inverse of the function $f\left( x \right)$ , means we can write it as;
$\because g = {f^{ - 1}}\left( x \right)$ ( given )
Therefore, $fog\left( x \right) = x$
$\Rightarrow f\left[ {g\left( x \right)} \right] = x$
Differentiate the above equation with respect to $x,$ we get;
Using the chain rule of differentiation;
\Rightarrow f'\left\\{ {g\left( x \right)} \right\\} \times g'\left( x \right) = 1
From the above equation the value of $g'\left( x \right)$ can be written as;
\Rightarrow g'\left( x \right) = \dfrac{1}{{f'\left\\{ {g\left( x \right)} \right\\}}}{\text{ }}......\left( 1 \right)
The value of $f'\left( x \right) = \dfrac{1}{{1 + {x^5}}}$ ( given )
Now, put the value of $f'\left( x \right) = \dfrac{1}{{1 + {x^5}}}$ in equation $\left( 1 \right)$ ;
Replace the argument $x{\text{ of }}f\left( x \right)$ with $g\left( x \right)$ , we get;
\Rightarrow g'\left( x \right) = \dfrac{1}{{\dfrac{1}{{1 + {{\left\\{ {g\left( x \right)} \right\\}}^5}}}}}
The above equation can also be written as;
\Rightarrow g'\left( x \right) = 1 + {\left\\{ {g\left( x \right)} \right\\}^5}

So, the correct answer is “Option 4”.

Note: For two functions to be inverse of each other, the inverse composition rule should be satisfied. These are the conditions for $f\left( x \right){\text{ and g}}\left( x \right)$ to be inverse of each other;
$\left( 1 \right)$ f\left\\{ {g\left( x \right)} \right\\} = x , for all the values of $x$ in the domain of $g\left( x \right)$ . $\left( 2 \right)$ g\left\\{ {f\left( x \right)} \right\\} = x , for all the values of $x$ in the domain of $f\left( x \right)$ . The domain of a composite function $f\left( {g\left( x \right)} \right)$ is the set of those values of $x$ in the domain of $g\left( x \right)$ for which $g\left( x \right)$ is in the domain of $f\left( x \right)$ .