The value of the derivative of the function g(x) at x = 1 will be 1/3. Then the correct option is A.
A function that may convert into another function is known as an inverse function or anti-function.
If the function is f(x) = sinx + 2x + 1.
Then the derivative of an inverse function g(x) of a function f(x) at a given value (a) will be
[tex]\rm g'(a) = \dfrac{1}{f'(g(a))}[/tex]
The derivative of f(x) will be
f'(x) = cos x + 2
Then the given value of a will be
a = 1
g(x) = f⁻¹(x)
put x = 0, then we have
f(0) = sin 0 + 2(0) + 1
f(0) = 1
Put x = 1, in the function f⁻¹(x). Then we have
f⁻¹(1) = 0
and
g(1) = 0
Then put a = 1, then we have
[tex]\rm g'(a) = \dfrac{1}{f'(g(a))}\\\\g'(1) = \dfrac{1}{f'(g(1))}\\\\g'(1) = \dfrac{1}{f'(0)}\\\\g'(1) = \dfrac{1}{\cos 0 + 2}\\\\g'(1) = \dfrac{1}{1+2}= \dfrac{1}{3}[/tex]
More about the inverse function link is given below.
https://brainly.com/question/2541698
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