A reliable method
Name the outer and inner functions. For a product keep both product-rule terms; for a quotient square the denominator.
Worked foundation example
If f(x)=x²(x+3), find f′(2). The product rule gives f′(x)=2x(x+3)+x². This simplifies to 3x²+6x. At x=2, the value is 24.
The common trap
The derivative of a product is not the product of derivatives. Omitting the inner derivative is the most common chain-rule error.
Connect the representations
Check a complicated derivative by expanding when possible or by estimating a nearby difference quotient. Domain restrictions still apply.
Explain before checking
Write the defining equation, state its assumptions, and attach units before calculating. After checking the numerical answer, compare every step with the worked reasoning. These short tasks build selected foundations; a complete AP free-response solution also needs written justification, correct notation and attention to the rubric.
Compare product and composition
For p(x)=x²(x+3), the product rule gives 2x(x+3)+x²; expanding first gives x³+3x², whose derivative is 3x²+6x. These agree. For h(x)=(x²+3)², the outer square surrounds the entire inner expression, so the chain rule gives 2(x²+3)·2x. A product has two visible multiplied functions; a composition places one function inside another.
Transfer and explain
Transfer: differentiate x(x+1)² using a product rule outside and chain rule inside. The derivative is (x+1)²+2x(x+1). At x=1 it equals 8. Check by expanding to x³+2x²+x.
If f(x)=x²(x+3), find f′(2).
- The product rule gives f′(x)=2x(x+3)+x².
- This simplifies to 3x²+6x.
- At x=2, the value is 24.
Compare the other answer choices
Keep only the second product-rule term.
Drop the x and factor 2 from the linear derivative term.
Keep only the first product-rule term.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
AP Calculus AB courseCalculus Volume 1: Fundamental Theorem of Calculus