Question

Let f and g be differentiable functions satisfying $g'\left( a \right) = 2$, $g\left( a \right) = b$ and $f \circ g = I$ (Identity function). Then $f'\left( b \right)$ is equal to
A. $\dfrac{1}{2}$
B. $\dfrac{2}{3}$
C. $2$
D. None of these

Answer 1

Hint : Here, the given question. We have to find the derivative or differentiated term of function. For this, first consider the given composite function which is an identity function then differentiating by using chain rule for differentiation. And to further simplify using a given condition we get the required differentiation value.

Complete step-by-step answer :
Differentiation can be defined as a derivative of a function with respect to an independent variable
Otherwise
The differentiation of a function is defined as the derivative or rate of change of a function. The function is said to be differentiable if the limit exists.
Let $y = f\left( x \right)$ be a function of. Then, the rate of change of “y” per unit change in “x” is given by $\dfrac{{dy}}{{dx}} = f'(x)$.
The Chain Rule is a formula for computing the derivative of the composition of two or more functions.
The chain rule expressed as $f'(x) = \dfrac{{dy}}{{dx}} = \dfrac{{dy}}{{du}} \cdot \dfrac{{du}}{{dx}}$
Consider the given composite function
$f \circ g = I$ (Identity function)
So,
$\Rightarrow \,\,f\left( {g\left( x \right)} \right) = x$
Differentiate with respect to x by using a chain rule of differentiation
$\Rightarrow \,\,f'\left( {g\left( x \right)} \right)g'\left( x \right) = \dfrac{d}{{dx}}\left( x \right)$
$\Rightarrow \,\,f'\left( {g\left( x \right)} \right)g'\left( x \right) = 1$
Put, $x = a$
$\Rightarrow \,\,f'\left( {g\left( a \right)} \right)g'\left( a \right) = 1$
Divide both side by $g'\left( a \right)$, then
$\Rightarrow \,\,f'\left( {g\left( a \right)} \right) = \dfrac{1}{{g'\left( a \right)}}$
Since, by given $g'\left( a \right) = 2$, $g\left( a \right) = b$, then
On simplification we get
$\therefore \,\,\,\,f'\left( b \right) = \dfrac{1}{2}$
Hence, it’s a required solution
Therefore, Option (A) is the correct answer.
So, the correct answer is “Option A”.

Note : ‘$f \circ g$’ is the composite function of $f\left( x \right)$ and $g\left( x \right)$. The composite function $f \circ g$ can be written as $f\left( {g\left( x \right)} \right)$ is read as “f of g of x” i.e., “the function g is the inner function of the outer function f” and remember the standard differentiation formula If the function is complex form we have to use the chain rule differentiation it makes easy to find out the differentiated term.