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