what is the definition of an inverse? in functions, two functions are inverses (we disregard the property of one-to-one: i.e. we restrict the domain so that the function is one-to-one, and we do this every period) if you compose them, then it gives you the indentity function..
let us consider a continuous function on a point by point basis.. what if the value of the function at x=a is 0, then what is the inverse of zero?
as in the example, if the derivatives are another function, you can observe that there are points on the blue sinosoidal curve (the derivative one) which approaches y=0, hence its inverse approaches infinity at these points..
besides, observe the original function and its inverse.. if you look at the points of intersection, the blue one is almost horizontal while the green one is almost vertical.. and what is the slope of the tangent line (or the derivative) if these were the situation? (this fact coincides with the previous paragraph..)
i hope you got what i mean.. Ü