Definition of the derivative
A rate of change needs two points, yet a speedometer shows a speed at one instant. How can a single point have a slope?
Where you are 3 inside
What this area is about
An average rate of change is the slope of a secant line through two points of a graph. Slide the second point toward the first and the secant slopes settle on one number, the slope of the curve at that point. That limit,
How these fit together
Definition. The derivative at a point defines f'(a) as the limit of the difference quotient and reads it as the slope of the tangent line and an instantaneous rate.
Procedure. Finding a derivative from the definition evaluates the same limit by hand with x left as a letter, giving a formula for f'(x) at every point at once.
The derivative as a function. The derivative as a function treats f' as a function with its own domain and graph, sketched from a graph of f or estimated from a table.
A path through it
The derivative at a point comes first. Finding a derivative from the definition comes next, and the derivative as a function last, since it builds on the formulas the other two produce.
Finish here and the shortcuts begin: the power rule, the product rule and the second derivative all rest on this definition.