Compare nearby function values
An average rate of change divides the change in output by the change in input. A derivative takes the limit of that comparison as the input interval shrinks, when the limit exists.
Connect the number to its meaning
On a graph, the derivative is the tangent slope. If the function describes position over time, the derivative describes velocity. Its units follow from output units divided by input units.
A mistake to avoid
The value f(a) and the derivative f′(a) answer different questions. Also, a graph can have points where no derivative exists, so do not assume every visible curve has a finite tangent slope everywhere.
Worked example
Find the slope of f(x) = x² at x = 3
For nonzero h, the average rate is ((3 + h)² − 9)/h = 6 + h. As h approaches zero, the rate approaches 6. Thus f′(3) = 6, while f(3) = 9 is the function’s height.