what does y'(0) mean for a function ?

Actually you are partially right. After differentiation you have frame an equation in which except dy/dx no term will be present in y, only x and constants will be present in the equation and you have to substitute x with 0.

In order to do so you will have to use various substitutions used by you in the methods you worked out for solition. Such as the value of y is most of the time given in the qustion only.