Pedagogy:Product rule for differentiation

These are pedagogical notes on the product rule for differentiation.

Pedagogical contexts

Course level	Goal	Coverage suggestions
high school calculus (e.g., AP Calculus, IB Calculus, or A levels)	Computational fluency with using product rule numerically and algebraically	Focus on the formula aspect for product of two functions. Relevant sections: product rule for differentiation#Statement with symbols, product rule for differentiation#Computational feasibility significance (provide examples and don't overemphasize the theoretical significance), product rule for differentiation#Examples (exclude sanity checks in first treatment except as asides).
single variable calculus class (college class or gifted/talented class) with partial focus on theory and proofs	Build on existing computational fluency and provide insight into subtleties and existential questions	Review formula aspects quickly, then cover a selection of these based on time constraints: multiple functions, one-sided versions, significance (qualitative/existential, computational feasibility, computational results), product rule for differentiation#Sanity checks. Exclude the compatibility checks as they may be too tricky to motivate, at least in the initial treatment.
single variable calculus class, review time	provide more insight into subtleties and existential questions	Review the rule while doing integration by parts (see product rule for differentiation#Reversal for integration) Cover more subtleties/aspects of the rule at opportune moments, e.g., when covering vertical tangents and vertical cusps, when discussing one-sided differentiability, when discussing piecewise functions. Cover the product rule when comparing with the product rule for differentiation of formal power series.
multivariable calculus class	cover product rule for partial differentiation, product rule for differentation of dot product, etc.	Emphasize key features of the product rule and which of them do and don't generalize to the multivariable calculus situation. At a later stage in the class, consider covering the proof of product rule for differentiation using chain rule for partial differentiation.
single variable calculus class for math majors, with strong focus on proofs	Teach how one would prove it, how one would think of it, and how it fits in with the other rules.	cover one or more proof ideas. Start off with proof of product rule for differentiation using difference quotients. Cover one or more of the topics: product rule for differentiation#Compatibility checks, multiple functions, one-sided versions, significance (qualitative/existential, computational feasibility, computational results) After introducing the natural logarithm, consider the proof of product rule for differentiation using logarithmic differentiation.

Error types

Erroneous rule for $(f\cdot g)'$	How to prevent this error
$f'\cdot g'$, i.e., the freshman product rule	explicitly warn against this rule, both as a strong injunction and providing clear explanations for why it is not true (the product rule for differentiation page has details on why the freshman product rule does not make sense). This can be overcome completely.
$f'\cdot g$ (or $f\cdot g'$), i.e., just one of the two terms of the product rule	this is rarer but harder to overcome completely as it may occur as a result of attention loss during computation and not due to a fundamental conceptual misunderstanding. Attention to the form of answers (i.e., to remembering that product rules involve the sum of two terms, so if the answer doesn't look like that, then you may have missed something) helps.