You can approximate the slope of the graph of a function for a given point value of x using the difference quotient over the interval . You let get closer and closer to and check if the difference quotient is approaching a certain limit value. If this is the case, then you have found the derivative, the required slope. The difference quotient is defined as:
When you divide by (met ) you are left with an expression that only depends on the value of as gets closer to . (Although the slope in the above graph is positive, can of course also be negative!)
This is the derivative for any given x.
The function is accordingly called the derivative function. You write it as .
This derivative represents the slope of the graph of the function for any given . It is also the slope of the tangent of the graph of at this value of .
The graph of is the graph of the slopes of .