Critical features of formal and informal reasoning in the case of the concept of derivative

Mathematical reasoning can be categorised into formal and informal ones on the basis of the type of used arguments. Both types of reasoning are needed in mathematics, but students usually tend to restrict their reasoning either to formal or informal mode only. Three examples collected and re-analysed from earlier studies reveal some critical features of this kind of restricted reasoning: A formal reasoning may easily become dependent on memory or authoritative references or it may become superficial imitation of algorithms. Informal reasoning, for one, may easily be based on intuitive conceptions, and its validity may be difficult to evaluate