given f(x)= (x^5)+2(x^3)+x-1if i have already found f(1)& f'(x) & f'(1), how do i find f-1'(1)?  f-1(x) denotes the inverse of f(x) and f'(x) is the derivative of f(x)

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to find the inverse of the function f(x) but it seems to be difficult to do it, hence you need to use the next relation between the derivative of function and inverse derivativeof function:


You need to find the derivative of the function such that:

`f'(x) = 5x^4 + 6x^2 + 1`

You need to find the derivative of the function at x=1, hence you need to substitute 1 for x in equation of derivative:

`f'(1) = 5 + 6 + 1 = 12`

Considering the relation `f'(x)*f^(-1)'(x)=1`  and substituting 1 for x yields:

`f'(1)*f^(-1)'(1)=1 =gt f^(-1)'(1)= 1/(f'(1))`

Substituting 12 for f'(1) yields:

`f^(-1)'(1)= 1/12`

Hence, evaluating the value of inverse derivative at x = 1 yields `f^(-1)'(1)= 1/12` .

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