C08 · Unit C2 Derivative foundations
Product, quotient and reciprocal trig rules
CoreReview editionNot yet human-reviewed
- Time
- About 45 minutes, untimed
- Course framework reference
- AP 2.8–2.10
- Builds on
- Part of
Progress on this lesson counts in every pathway listed.
Understand the idea
The product rule is (uv) minutes equals u minutes v+uv minutes: each factor contributes to the rate. The quotient rule is (u over v) minutes equals (u minutes v minus uv minutes)/v squared where v is not equal to 0. Dividing sine by cosine derives (tan x) minutes equals sec squared x. Similarly sec minutes equals sec tan, csc minutes equals minus csc cot and cot minutes equals minus csc squared on their domains.
Worked example
For f(x) equals x squared eˣ, f minutes equals 2xeˣ+x squared eˣ equals eˣ(x squared +2x). For g(x) equals (x+1)/(x minus 1), g minutes equals [(x minus 1) minus (x+1)]/(x minus 1) squared equals minus 2/(x minus 1) squared, x is not equal to 1.
Explore the mathematics
Product, quotient and reciprocal trig rules
Track the two changing factors in the derivative of x·eˣ.
Scroll sideways to see the full diagram.
Predict before continuing
Can the derivative of a product be found by multiplying derivatives?
Read the complete explanation
- Can the derivative of a product be found by multiplying derivatives?
- Changing x changes both factors; there are two first-order contributions.
- The product rule gives f′=u′v+uv′, not u′v′.
- At x=4, f′=272.9908. The omitted product of the two small changes vanishes in the limit.
For f(x)=x²eˣ, f′=2xeˣ+x²eˣ=eˣ(x²+2x). For g(x)=(x+1)/(x−1), g′=[(x−1)−(x+1)]/(x−1)²=−2/(x−1)², x≠1.
Discuss: explain what each label represents, then draw a related example with different values.
A common misconception
The derivative of a product is generally not the product of derivatives. Quotient-rule numerator order determines the sign.
Investigate and explain
Expand (x+1)(x squared minus 1), differentiate, and compare with product rule. Use equality of the results as an algebra check.
Identify one step in the worked example, name the mathematical fact that permits it, and state its domain conditions.