The derivative of a product is not the product of the derivatives. Checking x · x against 1 · 1 shows it: the answer is 2x, not 1.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The Product Rule
For f · g the derivative is f′g + fg′: differentiate each factor in turn while leaving the other alone.
The Quotient Rule
For f ÷ g the derivative is (f′g − fg′) ÷ g². The subtraction means the order genuinely matters.
Top first
The numerator begins with the derivative of the top. Starting with the bottom produces the answer with the wrong sign.
Sometimes you need neither
Simplify before differentiating where you can. (x³ + x) ÷ x is x² + 1, and the Power Rule finishes it in one step.
A way to remember
The product rule adds, and the quotient rule subtracts and divides by the bottom squared.
Products do not differentiate piecewise
The derivative of a product is not the product of the derivatives. Test it on x·x: the true derivative is 2x, not 1. The product rule exists precisely because the naive guess is wrong.
The product rule
(uv)′ = u′v + uv′. Differentiate each factor in turn, leaving the other alone, and add. The symmetric shape makes it easy to remember and easy to check.
The quotient rule
(u/v)′ = (u′v − uv′)/v². The minus sign means order matters here, unlike in the product rule. Getting the terms the wrong way round gives the negative of the right answer.
Simplify before differentiating
Many quotients are easier rewritten with a negative exponent and differentiated by the power rule. Choosing the easier route is legitimate — the quotient rule is not compulsory just because a fraction is present.
Step 2: Try It Yourself
Tap and try it out.
- Point(1, 1)
- Slope of the tangent3
Step 3: Watch an Example
One step at a time.
Watch Rosa Use the Product Rule
Rosa differentiates y = x² · (3x + 1).
- Step 1
She names the factors: f = x² and g = 3x + 1.
Step 4: Your Turn
Practice makes it stick.
The Check
Problem 1 of 2
y = x · x. Using the Product Rule, what is the derivative at x = 3?
The Simplification
Problem 2 of 2
y = (x³ + x) ÷ x simplifies to x² + 1. What is the derivative at x = 4?
Products and Quotients
1 of 8
y = x², so y′ = 2x. What is y′ at x = 6?
2 of 8
y = x² (3x + 1), so y′ = 9x² + 2x. What is y′ at x = 1?
3 of 8
y = x³, so y′ = 3x². What is y′ at x = 2?
4 of 8
y = 5x⁴, so y′ = 20x³. What is y′ at x = 1?
5 of 8
Is the derivative of a product the product of the derivatives? 1 for yes, 0 for no.
6 of 8
y = (x² + 3x) ÷ x simplifies to x + 3. What is the derivative?
7 of 8
Match each expression with the rule that fits best.
Tap a card on the left to start.
8 of 8
In the Quotient Rule numerator, which derivative comes first? 1 for the top, 2 for the bottom.
Step 5: Quick Check
Show what you know.
Question 1 of 2
y = x² (3x + 1), so y′ = 9x² + 2x. What is y′ at x = 2?
Question 2 of 2
What is the Product Rule?
What You Learned
- The derivative of a product is f′g + fg′, never f′g′.
- The derivative of a quotient is (f′g − fg′) ÷ g², with the top differentiated first.
- Simplifying first often removes the need for either rule.