AP Calculus AB/BC · Unit 2 · Topic 2.8
The Product Rule
Differentiate products without multiplying only the two derivatives.
1. Topic Focus
Define derivatives as limits, estimate slopes from representations, and establish the fundamental derivative rules.
This topic: Differentiate products without multiplying only the two derivatives.
2. Key Relationship
Read every symbol with its domain, direction, units, and hypotheses before applying the relationship.
3. Visual Connection
4. Worked Example
For x²sin x, the derivative is 2x sin x+x²cos x.
Write the governing relationship first, carry out the algebra cleanly, and finish with a sentence that answers the mathematical question.
5. Concept Development
When two differentiable factors both change, the product changes through two contributions. If \(P(x)=f(x)g(x)\), then
Differentiate one factor while holding the other at its current value, then reverse the roles and add. The order of the two terms may be switched, but each term must contain exactly one differentiated factor.
| Contribution | Changing factor | Held at its current value | Term |
|---|---|---|---|
| First | \(f\) | \(g\) | \(f'g\) |
| Second | \(g\) | \(f\) | \(fg'\) |
Why two terms appear. Start from the derivative definition and add and subtract \(f(x)g(x+h)\) in the numerator:
The last step uses continuity of differentiable functions, so \(g(x+h)\to g(x)\).
Why \(f'g'\) is not the rule. Write \(x^2=x\cdot x\). Multiplying the derivatives would give \(1\cdot1=1\), but the power rule gives \(2x\). The actual product rule gives \(1(x)+x(1)=2x\).
Using values from a table or graph. To find \((fg)'(a)\), four values may be needed:
Do not substitute only the two derivative values. Original function values describe the current sizes of the factors and are part of the rate.
Three or more factors. Differentiate one factor at a time while preserving every other factor:
The pattern extends to additional factors, producing one term per factor.
Choose an efficient representation. If all factors are short polynomials, expanding first may turn the problem into a simpler sum. If a factor is trigonometric, exponential, logarithmic, or an unspecified function, the product rule usually preserves structure and reduces algebra. Both valid routes must agree on their common domain.
Domain and structure check. A product is defined only where every factor is defined. The product rule does not remove restrictions from roots, logarithms, or reciprocals. Also distinguish multiplication from composition: \(f(x)g(x)\) uses the product rule, while \(f(g(x))\) requires the chain rule.
6. Detailed Worked Example and Error Check
Example 1: Algebraic times exponential. Let \(y=(x^2+1)e^x\). Label \(f=x^2+1\) and \(g=e^x\):
The first line displays the two product-rule contributions. Factoring afterward is optional.
Example 2: Polynomial times trigonometric.
The sine factor stays unchanged in the first term; the \(x^3\) factor stays unchanged in the second.
Example 3: Use a table of values. Suppose \(f(a)=2\), \(f'(a)=-1\), \(g(a)=5\), and \(g'(a)=3\). For \(H=fg\),
Multiplying only \(f'(a)g'(a)\) would incorrectly give \(-3\).
Example 4: Product rule and a tangent line. Let \(p(x)=(x^2+1)(x-2)\). At \(x=1\), \(p(1)=-2\), and
The tangent line is horizontal: \(y=-2\).
Example 5: Three changing factors. For \(F(x)=x^2e^x\sin x\),
There are three terms because each of the three factors takes one turn being differentiated.
Example 6: Product rates in context. A rectangle has changing length \(L(t)\) and width \(W(t)\), so \(A(t)=L(t)W(t)\). At one instant, suppose
with lengths in meters and rates in meters per second. Then
The area is increasing even though the width is decreasing because the length contribution is larger.
AP error check. Do not multiply only the derivatives, differentiate both factors in the same term, omit one contribution, replace table values of \(f\) and \(g\) with their derivatives, or use the product rule for a composition.
7. AP Reasoning Routine
Identify the function structure, state the applicable rule, preserve notation and units, and check differentiability before interpreting a derivative.
- Identify the representation and requested quantity.
- State the rule or theorem and verify its conditions.
- Keep exact values until the final requested approximation.
- Interpret sign, units, interval, and context.
Use the product rule and preserve structure until all terms are present.
(a) Differentiate \(y=(x^2-3)\cos x\).
(b) If \(f(a)=4\), \(f'(a)=2\), \(g(a)=-1\), and \(g'(a)=5\), find \((fg)'(a)\).
(c) Use \(x^2=x\cdot x\) to explain why \((fg)'=f'g'\) cannot be correct in general.
(d) Differentiate \(q(x)=(2x+1)(x^3-4)\).
(e) Find the tangent line to \(p(x)=(x^2+1)(x-2)\) at \(x=1\).
(f) Differentiate \(R(x)=x^2e^x\sin x\).
(g) A rectangle has \(L=10\) cm, \(W=6\) cm, \(dL/dt=0.4\) cm/s, and \(dW/dt=-0.2\) cm/s. Find and interpret \(dA/dt\).
(h) Differentiate \((x+2)(x^2-3x+1)\) both by expanding first and by using the product rule, then verify that the results agree.
Check the solution
(a) \(y'=2x\cos x-(x^2-3)\sin x\).
(b) \((fg)'(a)=2(-1)+4(5)=18\).
(c) The incorrect rule gives \(1\cdot1=1\), while the actual derivative of \(x^2\) is \(2x\). The product rule correctly gives \(1(x)+x(1)=2x\).
(d) \(q'(x)=2(x^3-4)+(2x+1)(3x^2)\).
(e) \(p(1)=-2\) and \(p'(1)=0\), so the tangent line is \(y=-2\).
(f) \(R'(x)=2xe^x\sin x+x^2e^x\sin x+x^2e^x\cos x\).
(g) \(A'=L'W+LW'=(0.4)(6)+(10)(-0.2)=0.4\) cm²/s. At that instant, area is increasing at 0.4 square centimeters per second.
(h) Expanding gives \(x^3-x^2-5x+2\), whose derivative is \(3x^2-2x-5\). The product rule gives \((x^2-3x+1)+(x+2)(2x-3)\), which simplifies to the same result.