# Pedagogy:Product rule for differentiation

These are pedagogical notes on the product rule for differentiation.

## Pedagogical contexts

### Course 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 Use School:Product rule for differentiation
Long version: 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).
Do more worked examples or assign worked examples as homework. The examples should cover all the function types that the students are familiar with, and as they see new functions, they should see at least one application of the product rule to each new type of function.
single variable calculus class (college class or gifted/talented class) with partial focus on understanding theory (but not coming up with proofs of theorems by oneself) Build on existing computational fluency and provide insight into subtleties and existential questions, i.e., into what the results tell us qualitatively. Use College:Product rule for differentiation
Long version: 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), sanity checks. Exclude the compatibility checks as they may be too tricky to motivate, at least in the initial treatment.
Exclude proof from the initial treatment. If you wish to include some proof, consider a proof of differentiation is linear, which is easier to understand and covers the same ideas.
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 (including coming up with proofs oneself) Teach how one would prove it, how one would think of it, and how it fits in with the other rules. Use Mathmajor:Product rule for differentiation
Long version: 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: 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.

### Independent learners

For self-directed learning, consider the following:

• First follow the coverage suggestions for the "high school calculus" version.
• Then, if you are highly interested in proof-based mathematics, skip directly to the "single variable calculus for math majors, with strong focus on proofs" version.
• Otherwise, continue to the more basic "single variable calculus (college class)" version, and if interested further, fill in the proof-related gaps later.

## Error types

Erroneous rule for $(f \cdot g)'$ Notes on 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.
differentiation by parts, trying to do integration by parts for differentiating a product This is a rare error, and is likely to occur only if the person has been doing too much integration in the recent past. More attentiveness is the best bet against this.