CCD.DVDD: The derivative of $f$ at $a$ is a number, denoted $f'(a)$, which is defined as  $$f'(a) = \lim_{x \rightarrow a} \frac {f(x)-f(a)}{x-a}$$ or  $$f'(a) = \lim_{h \rightarrow 0} \frac {f(a+h)-f(a)}{h}$$.
The ratios used in the limit expressions can be interpreted as
  1. slopes of secant lines  from a fixed point to a second varying point on a curve that is the graph of a function,
  2. average velocities using a fixed reference time for an object moving on a straight line, or
  3. average rates of change of a dependent variable with respect to an independent variable with one fixed value for the independent variable.
The derivative can also be understood as the magnification factor of the best linear approximating function at a specific value for the controlling variable.

On the mapping diagram the derivative can be visualized using focus points for the average rates that converge to a single focus point for the best linear approximating function.
The derivative can also be visualized as a vector on the target of the mapping diagram.