Product Rule
Key Formula
dxd[f(x)⋅g(x)]=f′(x)⋅g(x)+f(x)⋅g′(x)
Where:
- f(x) = The first function
- g(x) = The second function
- f′(x) = The derivative of the first function
- g′(x) = The derivative of the second function
Worked Example
Problem: Find the derivative of h(x) = x² · sin(x).
Step 1: Identify the two functions being multiplied.
f(x)=x2,g(x)=sin(x)
Step 2: Find the derivative of each function separately.
f′(x)=2x,g′(x)=cos(x)
Step 3: Apply the product rule: multiply f'(x) by g(x), then add f(x) times g'(x).
h′(x)=f′(x)⋅g(x)+f(x)⋅g′(x)=2x⋅sin(x)+x2⋅cos(x)
Step 4: Write the final simplified result.
h′(x)=2xsin(x)+x2cos(x)
Answer: h'(x) = 2x sin(x) + x² cos(x)
Another Example
This example uses two polynomials, which lets you verify the product rule by expanding first and differentiating directly—a useful check that both methods give the same result.
Problem: Find the derivative of y = (3x + 1)(x³ − 2x).
Step 1: Identify the two polynomial factors.
f(x)=3x+1,g(x)=x3−2x
Step 2: Differentiate each factor.
f′(x)=3,g′(x)=3x2−2
Step 3: Apply the product rule.
y′=3(x3−2x)+(3x+1)(3x2−2)
Step 4: Expand each part and combine like terms.
y′=3x3−6x+9x3−6x+3x2−2=12x3+3x2−12x−2
Step 5: You can verify this by expanding the original product first: (3x + 1)(x³ − 2x) = 3x⁴ + x³ − 6x² − 2x, then differentiating term by term to get the same answer.
dxd(3x4+x3−6x2−2x)=12x3+3x2−12x−2✓
Answer: y' = 12x³ + 3x² − 12x − 2
Frequently Asked Questions
Why can't you just multiply the derivatives of the two functions?
Because differentiation does not distribute over multiplication in that way. The derivative of f(x)·g(x) is NOT f'(x)·g'(x). The product rule accounts for the fact that both functions are changing simultaneously; you need to consider each function's rate of change while holding the other function in place. A quick counterexample: the derivative of x·x = x² is 2x, but 1·1 = 1, which is clearly wrong.
When do you use the product rule versus the chain rule?
Use the product rule when two functions are multiplied together, like f(x)·g(x). Use the chain rule when one function is composed inside another, like f(g(x)). For example, x²·sin(x) needs the product rule, while sin(x²) needs the chain rule. Some problems require both rules at once.
Does the product rule work for more than two functions?
Yes. For three functions f·g·h, the derivative is f'gh + fg'h + fgh'. The pattern extends to any number of factors: you differentiate one factor at a time while leaving the others unchanged, then add all those terms together.
Product Rule vs. Quotient Rule
| Product Rule | Quotient Rule | |
|---|---|---|
| Definition | Derivative of f(x) · g(x) | Derivative of f(x) / g(x) |
| Formula | f'g + fg' | (f'g − fg') / g² |
| When to use | Two functions multiplied together | One function divided by another |
| Sign between terms | Addition (+) | Subtraction (−), so order matters |
| Denominator | None | g(x) squared |
Why It Matters
The product rule is one of the first differentiation rules you learn in calculus and appears constantly in AP Calculus, university-level courses, and physics. Many real-world quantities—such as force (mass × acceleration) or revenue (price × quantity)—are naturally expressed as products of changing functions. Without the product rule, you would be unable to differentiate most functions that arise in applications.
Common Mistakes
Mistake: Multiplying the two derivatives together: writing the derivative of f·g as f'·g'.
Correction: The correct formula is f'g + fg'. Differentiation does not distribute over multiplication. A quick check with f(x) = g(x) = x shows the error: (x·x)' = 2x, not 1·1 = 1.
Mistake: Forgetting one of the two terms in the product rule, typically the second term fg'.
Correction: Always write both terms explicitly before simplifying. A helpful mnemonic is "first times derivative of the second, plus second times derivative of the first," and make sure you see two separate addends.
Related Terms
- Derivative — The broader concept the product rule computes
- Derivative Rules — Collection of rules including the product rule
- Function — The objects being multiplied and differentiated
- Formula — The product rule is a key differentiation formula
- Product — The multiplication operation the rule addresses
- Chain Rule — Differentiation rule for composite functions
- Quotient Rule — Analogous rule for dividing two functions
- Power Rule — Basic rule often used within the product rule