Question

If g is the inverse function of \[f\] and \[f'(x)=\dfrac{1}{1+{{x}^{n}}}\], then \[g'(x)\] is equal to.
(a) \[1+{{[g(x)]}^{n}}\]
(b) \[1-g(x)\]
(c) \[1+g(x)\]
(d) \[-g{{(x)}^{n}}\]

Answer 1

Hint: We will use the concept of composition of functions to solve this question. Composite function is the combination of two given functions in which we can apply the first function as well as get the answer as well as after that, we can use this answer into the second function. Composite function is the application of one function to the result of another.

Complete step-by-step answer:
Before proceeding with the question we should understand the concept of composition of functions.
The composition of a function is an operation where two functions say f and g generate a new function say h in such a way that h(x) is equal to g(f(x)). It means here function g is applied to the function of x. So, basically, a function is applied to the result of another function.
It is mentioned in the question that g is the inverse function of \[f\]. So using this information we get,
\[\Rightarrow g(x)={{f}^{-1}}(x).........(1)\]
Multiplying both sides of the equation (1) by f we get f of g in the left hand side of the equation (1). So doing this we get,
\[\Rightarrow f\circ g(x)=f\circ {{f}^{-1}}(x).........(2)\]
Now solving and rearranging equation (3) we get,
\[\Rightarrow f(g(x))=x.........(3)\]
Now differentiating the both sides of the equation (3) we get,
\[\Rightarrow \dfrac{df(g(x))}{dx}=\dfrac{d}{dx}(x).........(4)\]
So now solving and simplifying equation (4) we get,
\[\Rightarrow f'(g(x)).g'(x)=1.........(5)\]
Now \[f'(x)=\dfrac{1}{1+{{x}^{n}}}\] is mentioned in the question so substituting g(x) in it and then in equation (5) we get,
\[\Rightarrow \dfrac{1}{1+{{[g(x)]}^{n}}}\times g'(x)=1.........(6)\]
So now solving equation (6) and rearranging we get,
\[\Rightarrow g'(x)=1+{{[g(x)]}^{n}}.........(7)\]
So from this in equation (7) we get \[1+{{[g(x)]}^{n}}\] as the answer. And hence option (a) is the correct answer.

Note: Remembering the concept of composition of functions is the key here. Here g is the inverse of f so from this we will find f(g(x). We in a hurry can make a mistake in differentiating equation (4) so we need to be careful while doing this step.