How determine derivative of inverse of f(x)=e^x+x^2+1 in point e+2?

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The inverse function is a surjective function, hence, by surjective function definition, the following equation needs to have one solution, such that:

`e^x_0+x_0^2+1 = e + 2 => e^x_0+x_0^2 = e + 1`

Replacing 1 for `x_0` yields:

`e^1 + 1^2 = e + 1 ` valid

Hence, the only solution to the equation `e^x_0+x_0^2+1 = e + 2` is `x_0 = 1` .

You need to evaluate the derivative of inverse function at x = e + 2, hence, you need to use the relation between derivative of function and derivative of its inverse, such that:

`(f^(-1)(e+2)) = 1/(f'(1))`

`f'(x) = e^x + 2x => f'(1) = e + 2`

`(f^(-1)(e+2)) = 1/(e + 2)`

**Hence, evaluating the derivative of inverse function, at `x = e ` `+ 2` , yields **`(f^(-1)(e+2)) = 1/(e + 2).`

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