From Derivative to Proportionality

Step 1: Description of Bottleneck

Students have difficulties to use derivative to calculate the change of function value. They believe the derivative is a magic transformation of one formula into another, e.g. x² into 2x. In case they know, that "derivative=slope", they do not know what exactly the slope means, because they do not calculate "change of function value=slope*change of argument value".

Desired learning outcome: Students should understand that the change of function value is proportional to the change of argument value, and the derivative is exactly the proportionality factor. Students should be able to construct a triangle on the graph of a function to visually represent changes in function values and the corresponding slope (rate of change).

Concrete problems may be, for example:

• A) Given that f(1)=2 and f'(1)=3 students should be able to make an estimation of f(1,01) as f(1)+f'(1)*0,01=2.003
• B) Given the entities of function and entities of argument, decide on the entities of derivative. E.g. fuel price: Argument x is in Litre, function f(x) is in Euro, follows that the derivative f'(x) is in Euro/Litre
• C) Given the text, which describes the rate of change, deduce a differential equation. E.g. the population growth: Newborns are proportional to the population number leads to the equation dN/dt=C*N

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

1. 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)
2. 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.
3. He can work with augmenting (moving to the right on the graph) and reducing (moving to the left) value of argument, and he can split the change of the value in multiple steps if he realizes that the step is too big.
4. 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.

People interested in this Bottleneck

Inna Mikhailova