We have seen how to define the derivative of a function y=f(x) as point a in the domain of f. It is the limit of the difference quotient of f at a as h approaches 0 :
Remember that the difference quotient is the slope of the secant line connecting the points (a,f(a)) and (a+h,f(a+h)) on the graph of f. The derivative is the slope of the tangent line to the graph of f at (a,f(a)).
We can use this to define a function, also called the derivative of f, which tells us the slope of the tangent line to the graph of f at (x,f(x)). We define it at a point x by the same limit:
For simple functions f we can carry out this calculation explicitly:
First, we simplify the difference quotient as much as possible;
second, we allow 
in the simplified quotient to find
the limit. Today, we'll work through several examples.