Cogito
Calculus · Chapter 4 · Lesson 2
The Product and Quotient Rules
Products do not differentiate one piece at a time.
12 problems · about 22 minutes · FUN-3.B
What this lesson teaches
The student differentiates products and quotients of functions.
- 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.
Warm Up
Straightforward practice. Get the method working first.
5 problemsy = x² (3x + 1), so y′ = 9x² + 2x. What is y′ at x = 2?
Answer 40
Why 40.
What is the Product Rule?
Answer f′g + fg′
Why f′g + fg′.
y = x², so y′ = 2x. What is y′ at x = 6?
Answer 12
Why 2 × 6.
y = x² (3x + 1), so y′ = 9x² + 2x. What is y′ at x = 1?
Answer 11
Why 9 + 2.
y = x³, so y′ = 3x². What is y′ at x = 2?
Answer 12
Why 3 × 4.
Build It Up
The same ideas with more to keep track of.
3 problemsy = 5x⁴, so y′ = 20x³. What is y′ at x = 1?
Answer 20
Why 20 × 1.
Is the derivative of a product the product of the derivatives? 1 for yes, 0 for no.
Answer 0
Why Test it on x times x.
y = (x² + 3x) ÷ x simplifies to x + 3. What is the derivative?
Answer 1
Why The slope of a line.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsMatch each expression with the rule that fits best.
Answer x² times (x + 1) → Product Rule; (x + 1) divided by (x − 1) → Quotient Rule; x⁵ → Power Rule
Why A single power needs no product or quotient rule at all.
In the Quotient Rule numerator, which derivative comes first? 1 for the top, 2 for the bottom.
Answer 1
Why Starting with the bottom flips the sign.
The Check: y = x · x. Using the Product Rule, what is the derivative at x = 3?
Answer 6
Why 6.
The Simplification: y = (x³ + x) ÷ x simplifies to x² + 1. What is the derivative at x = 4?
Answer 8
Why 8.