Key takeaways
- Product rule: (f × g)′ = f′ × g + f × g′.
- Say it as: derivative of the first times the second, plus the first times the derivative of the second.
- The #1 mistake is multiplying the two derivatives: (fg)′ is not f′g′.
- The order of the two terms doesn’t matter, because you add them.
In this guide
What is the product rule?
A derivative tells you how fast something is changing. When a quantity is made by multiplying two changing things, both parts add to the change.
Think of a rectangle whose width and height are both growing. The area grows because the width gets longer and because the height gets taller. The product rule adds those two effects: “first changes while second stays” plus “first stays while second changes”.
Why does the product rule matter?
Many real formulas are products: revenue = price × quantity, power = voltage × current, and physics formulas like t × e^(−t). When both parts change, you need the product rule to get the rate right.
Multiplying the derivatives can be badly wrong. For x² × x³ at x = 2, the true slope is 80, but f′ × g′ gives only 48.
The formula
| Symbol | What it means | Example (x² × x³) |
|---|---|---|
| f | The first function | x² |
| g | The second function | x³ |
| f′ | Derivative of the first | 2x |
| g′ | Derivative of the second | 3x² |
| f′g + fg′ | The product rule result | 2x⁴ + 3x⁴ = 5x⁴ |
This calculator uses f(x) = xᵃ and g(x) = xᵇ and gives the slope at your chosen x. Because xᵃ × xᵇ = xᵃ⁺ᵇ, you can check it with the power rule: the answer always equals (a + b)xᵃ⁺ᵇ⁻¹.
How to use the product rule step by step
- 1Name the two parts. Call them f and g.
- 2Differentiate each one. Find f′ and g′ separately.
- 3Multiply f′ by g. Keep g exactly as it is.
- 4Multiply f by g′. Keep f exactly as it is.
- 5Add and simplify. The answer is the sum of steps 3 and 4.
Now try it yourself. Enter the two powers and a value of x in the calculator below to see both terms added together.
Try it yourself
Pre-filled with the example — change any value.
Common mistakes to avoid
- Dropping one of the two terms. There are always two parts added together.
- Confusing it with the quotient rule. For division, the rule has a minus sign and a squared bottom. See the quotient rule calculator.
- Forgetting the chain rule inside. In
x² × sin(3x), the derivative of sin(3x) is 3cos(3x). See the chain rule calculator.
Tips and tricks
- Three functions: differentiate one at a time and keep the others:
(fgh)′ = f′gh + fg′h + fgh′. - Factor at the end.
d/dx [x eˣ] = eˣ + x eˣ = eˣ(1 + x). Factored answers are easier to use. - Backwards: run the product rule in reverse and you get integration by parts.
Where you'll use it in real life
- Business: revenue = price × number sold, when both change over time.
- Electronics: power = voltage × current in a changing circuit.
- Physics: motion formulas like
t × e^(−t)for a damped spring. - Geometry: the rate a rectangle’s area grows when both sides grow.
Quick summary
To differentiate f × g, find f′g + fg′: derivative of the first times the second, plus the first times the derivative of the second. Never just multiply the two derivatives. Use the calculator above to check xᵃ × xᵇ at any point, with every step shown.
Worked example
Your homework asks you to differentiate x² × x³ with the product rule and find the slope at x = 2, then check it against the power rule on x⁵. Here f = x² (a = 2) and g = x³ (b = 3).
- 01f(x) = x^2, g(x) = x^3
- 02f'(x) = 2x^1, g'(x) = 3x^2
- 03Product Rule: d/dx[f·g] = f'·g + f·g'
- 04= 2x^1 · x^3 + x^2 · 3x^2
- 05= 2x^4 + 3x^4 = 5x^4
- 06At x = 2: f'(2) = 4, g(2) = 8
- 07f(2) = 4, g'(2) = 12
- 08= 4 × 8 + 4 × 12 = 80
These numbers are pre-filled in the calculator above.
Frequently asked questions
What is the product rule formula?
If y = f × g, then y′ = f′g + fg′. Differentiate one part at a time, keep the other, and add the two results.
Does the order matter in the product rule?
No. Because you add the two terms, f′g + fg′ is the same as fg′ + f′g. Order only matters in the quotient rule.
How do I use the product rule with three functions?
Differentiate one function at a time and keep the others: (fgh)′ = f′gh + fg′h + fgh′.
What is the derivative of x times e^x?
f = x and g = eˣ, so f′ = 1 and g′ = eˣ. The derivative is eˣ + x eˣ, which is eˣ(1 + x).
