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Cogito

Calculus · Chapter 4 · Lesson 2

Practice Worksheet

The Product and Quotient Rules

Products do not differentiate one piece at a time.

12 problems · about 22 minutes · FUN-3.B

NameDate

Figure — use these to answer the problems

x · x² is x³. Its derivative is 3x², which the Product Rule also produces.
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y = 1x³ + 0x + 0
  • Point(1, 1)
  • Slope of the tangent3
a1
Point of tangency1
Figure 1

Warm Up

Straightforward practice. Get the method working first.

5 problems

  1. y = x² (3x + 1), so y′ = 9x² + 2x. What is y′ at x = 2?

    Answer

  2. What is the Product Rule?

    • f′g + fg′
    • f′g′

  3. y = x², so y′ = 2x. What is y′ at x = 6?

    Answer

  4. y = x² (3x + 1), so y′ = 9x² + 2x. What is y′ at x = 1?

    Answer

  5. y = x³, so y′ = 3x². What is y′ at x = 2?

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. y = 5x⁴, so y′ = 20x³. What is y′ at x = 1?

    Answer

  2. Is the derivative of a product the product of the derivatives? 1 for yes, 0 for no.

    Answer

  3. y = (x² + 3x) ÷ x simplifies to x + 3. What is the derivative?

    Answer

Stretch Yourself

Mixed problems. Work out what kind of question it is before you start.

4 problems

  1. Match each expression with the rule that fits best.

    Draw a line from each item on the left to its match on the right.

    • x² times (x + 1)
    • (x + 1) divided by (x − 1)
    • x⁵
    • Product Rule
    • Quotient Rule
    • Power Rule

  2. In the Quotient Rule numerator, which derivative comes first? 1 for the top, 2 for the bottom.

    Answer

  3. The Check

    y = x · x. Using the Product Rule, what is the derivative at x = 3?

    Answer

  4. The Simplification

    y = (x³ + x) ÷ x simplifies to x² + 1. What is the derivative at x = 4?

    Answer