I suspect that you have misunderstood the problem. To find the inverse of `y=x^3+x` you would exchange x and y and then solve for y; so solve `y^3+y=x` for y in terms of x. One way involves a substitution using the hyperbolic sin (sinh).
The solution (from an online calculator) is:
I suspect you were actually asked to find something like the value of `(f^(-1))'(2)` (find the derivative of the inverse function at a given point.) This you can do without calculating the inverse of the function using the derivative of an inverse function theorem from calculus.
If f is differentiable on some interval and f is invertible with inverse function g, then `g'(x)=1/(f'(g(x)))`
To find `(f^(-1))'(2)` we get `(f^(-1))'(2)=1/(f'(f^(-1)(2)))`
Since f(1)=2, we know that `g(2)=f^(-1)(2)=1` so we get
`f'(x)=3x^2+1==>f'(x)=4` so finally
** The solution above is incorrect. It is easily checked by graphing: the graph of the inverse is a reflection of the function over the line y=x.
Further, `f'(x)*(f^(-1))'(x) !=1` in general. (Try `y=x^2, x>=0` and its inverse `y=sqrt(x)` . However, if (a,b) is on f and (b,a) is on its inverse, then the slope of the tangent line on f at a is the reciprocal of teh slope of the tangent line at b on the inverse.
You should use the following relation between derivative of a function and its inverse, such that:
`f'(x)*(f^(-1)(x))' = 1`
Hence, you may find `(f^(-1)(x))'` dividing the equation above by `f'(x),` such that:
`(f^(-1)(x))' = 1/(f'(x))`
You need to differentiate the function `f(x)` with respect to `x` , such that:
`f'(x) = (x^3+x)' => f'(x) = 3x^2 + 1`
`(f^(-1)(x))' = 1/(3x^2 + 1)`
Hence, evaluating the derivative of the inverse function yields `(f^(-1)(x))' = 1/(3x^2 + 1).`