A function y = f(x) has an inverse if one and only one value of x gives a particular value of y. For example take the function f(x) = x^2. Here, there are two values of x, example 3 and -3, that give a unique value of y, 9 in for the values given. It is not possible here to find the inverse function f^-1(x) of f(x).
The derivative of a function f'(x) can be negative, positive or zero or all of these for different values of x. For a function to have an inverse or to be invertible it is essential that its derivative have values of only one sign. The derivative should either have only positive values or only negative values. Else, it is not possible to find the inverse.
If a function is invertible, the derivative of the function has either only positive values or only negative values.