What the Product Rule Is For
The power rule works beautifully when you have a single power of . But what happens when your function is the product of two functions, such as ?
You cannot simply differentiate each factor and multiply the results. Instead, the product rule accounts for the fact that both factors are changing as changes.
If one factor is and the other is , the product rule tells us exactly how the rates of change of both factors contribute to the derivative.
The Formula
If and are differentiable functions, then:
In words: differentiate the first function and multiply by the second, then add the first function multiplied by the derivative of the second.
A useful way to remember the pattern is: first derivative times second, plus first times second derivative.
Why There Are Two Terms
Suppose a product represents the area of a rectangle: . If both the width and height change, the area changes for two separate reasons:
- The width changes while the height is held at its current value.
- The height changes while the width is held at its current value.
The product rule captures both effects. That is why the derivative has two terms rather than just one.
The first term describes the change coming from the first factor. The second term describes the change coming from the second factor.
A Step-by-Step Method
- Identify the two factors. Decide which expression is the first function and which is the second function.
- Differentiate the first factor.
- Multiply by the original second factor. Do not differentiate the second factor in this term.
- Add the original first factor multiplied by the derivative of the second factor.
- Simplify, if appropriate.
Worked Examples
Differentiate .
Differentiate .
Differentiate . This example is important because the product rule and chain rule often appear together on AP Calculus questions.
Differentiate .
When You Do—and Don't—Need the Product Rule
Not every expression containing multiplication requires the product rule. Before using it, ask whether the expression can be rewritten more simply.
Example: A Constant Multiple
Consider . Although this contains multiplication, is a constant. You can simply use the constant multiple rule: .
Example: A Product of Two Functions
Now consider . Both factors depend on , so the product rule is appropriate.
Common Mistakes
A very common mistake is to write instead of using the product rule. The derivative of a product is not generally the product of the derivatives.
In each product-rule term, one factor is differentiated while the other factor remains unchanged. Do not differentiate both factors in the same term.
The product rule always has two contributions: derivative of the first times the second, plus the first times derivative of the second.
If an expression can easily be rewritten as a single power or simplified algebraically, doing so may be faster. For example, can be rewritten before differentiating. Always look for a simpler form when it is obvious.
Product Rule + Chain Rule: A Key AP Calculus Skill
On AP Calculus questions, the product rule rarely appears completely by itself. You may need to use it together with the chain rule.
For example, if , the product rule handles the multiplication, while the chain rule handles the derivative of the trigonometric factor.
The important strategy is to recognize that different rules can be nested inside one another. Use the product rule for the outer multiplication, then use whatever derivative rule is required for each individual factor.
Practice Problems
Differentiate .
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Differentiate .
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Differentiate .
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Differentiate .
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Quick Product Rule Checklist
- Do I have a product of two functions?
- What is my first factor?
- What is my second factor?
- What is the derivative of the first factor?
- What is the derivative of the second factor?
- Did I keep the undifferentiated factor in each term?
- Did I include the plus sign between the two terms?
- Do either of the individual derivatives require the chain rule?
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