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Derivative of the inverse function g at a point, where g(f(x)) = x

This type of question, asked 3 times: 25 Jun 2022, 4 Apr 2024, 5 Apr 2024. All 3 questions below, oldest first.

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  1. Question 1

    25 Jun 2022

    66
    Let f:R→Rf:R \rightarrow R be defined as f(x)=x3+x−5f(x) =x^{3}+ x - 5. If g(x)g(x) is a function such that f(g(x))=xf(g(x)) = x, ∀x∈R\forall x \in R, then gg ' (63) is equal to

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  2. Question 2

    4 Apr 2024

    79
    Let f(x)=x5+2ex/4f(x) =x^{5}+ 2e^{x/4} for all x∈Rx \in R. Consider a function g(x)g(x) such that (gof)(x)=x(gof)(x) = x for all x∈Rx \in R. Then the value of 8g′(2)8g^{'}(2) is:

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  3. Question 3

    5 Apr 2024

    77
    Let f(x)=x5+2x3+3x+1,x∈Rf(x) =x^{5}+ 2x^{3}+ 3x + 1,x \in R, and g(x)g(x) be a function such that g(f(x))=xg(f(x)) = x for all x∈Rx \in R. Then g(7)g′(7)\frac{g(7)}{g^{'}(7)} is equal to:

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