Editing Derivative (section)

=== Step 2: Description of mental tasks needed to overcome the bottleneck ===

# While thinking about derivative, a mathematician switches his mind  from "static" formula  story (algebra) to "dynamic" function  story (analysis).  He clearly understands that the story is not about calculating one value. It is about  how function value changes  if argument value changes (see a related bottleneck [[Function]]) 
# An expert can mask from his mind  the fact, that the derivative should be calculated as a limit.   He works with quotient "Change of function value" over "Change of argument value"  if this is useful to deduce needed information. E.g. if we are only interested in  entities or if  the formula f(x) is not given, but only numerical values. He deduces the needed values from the given graph of function by drawing a triangle in order to represent changes in values and the slope between two points.
# He can  work with augmenting (moving to the right on the graph) and reducing (moving to the left)  value of the argument, and he can split the change of the value in multiple steps if he realizes that the step is too big.
# While doing so,  he is not afraid of being unprecise, because he knows, that the approximation is locally allowed and can estimate the approximation error.