trouble with $f'$ I'm having a lot of trouble understanding the core concept of finding $f\,'(a)$ or any number in place of $a$. I do not understand what finding $f\,'$ means to begin with, or how to solve them. If somebody could give any information i would be appreciative. 

$f'(x)$ is the derivative of $f$ at $x$.
Other similar meanings are
rate of change
slope
grade (of an incline)
pitch, etc.
You know that slope of a line is the ratio of rise over run.
Well the graph of a function may not be a straight line but we can still find something that plays the same role as slope of a line. 
Here is how, at a point  with $x=a$ on the graph  draw its tangent line. The slope of this line is the derivative of $f$ at $a$. 
