Saman,

First of all I know that y = x³ + x has an inverse since it is an increasing function. I know this because the sign of the derivative is positive for every real number x. My guess is that this comes from a calculus problem here you are to find the derivative of the inverse at some point. If so then you don't need an explicit expression for the inverse. If you really want to find the inverse interchange the roles of x and y and write x = y³ + y . Rewrite this expression as

y³ + y - x = 0

and solve for y in terms of x. To do this you can use the method due to Tartaglia and published in 1545 by Girolamo Cardano. You can find description of this technique in the answer we gave to a previous student.

Harley