JEE MAIN - Mathematics (2014 (Offline) - No. 10)
If $$g$$ is the inverse of a function $$f$$ and $$f'\left( x \right) = {1 \over {1 + {x^5}}},$$ then $$g'\left( x \right)$$ is equal to:
$${1 \over {1 + {{\left\{ {g\left( x \right)} \right\}}^5}}}$$
$$1 + {\left\{ {g\left( x \right)} \right\}^5}$$
$$1 + {x^5}$$
$$5{x^4}$$
Explanation
Since $$f(x)$$ and $$g(x)$$ are inverse of each other
$$\therefore$$ $$g'\left( {f\left( x \right)} \right) = {1 \over {f'\left( x \right)}}$$
$$ \Rightarrow g'\left( {f\left( x \right)} \right) = 1 + {x^5}$$
$$\left( \, \right.$$ As $$\,f'\left( x \right) = {1 \over {1 + {x^5}}}$$ $$\left. \, \right)$$
Here $$x=g(y)$$
$$\therefore$$ $$g'\left( y \right) = 1 + \left\{ {g\left( y \right)} \right\}$$
$$ \Rightarrow g'\left( x \right) = 1 + \left\{ {g\left( x \right)} \right\}$$
$$\therefore$$ $$g'\left( {f\left( x \right)} \right) = {1 \over {f'\left( x \right)}}$$
$$ \Rightarrow g'\left( {f\left( x \right)} \right) = 1 + {x^5}$$
$$\left( \, \right.$$ As $$\,f'\left( x \right) = {1 \over {1 + {x^5}}}$$ $$\left. \, \right)$$
Here $$x=g(y)$$
$$\therefore$$ $$g'\left( y \right) = 1 + \left\{ {g\left( y \right)} \right\}$$
$$ \Rightarrow g'\left( x \right) = 1 + \left\{ {g\left( x \right)} \right\}$$
Comments (0)
