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Cogito

AP Calculus AB · Chapter 3 · Lesson 2

Practice Worksheet

The Chain Rule

Rates multiply along a chain.

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

NameDate

Figure — use these to answer the problems

Compare the steepness of x² with (3x)². Squeezing the inside multiplies the slope, which the Chain Rule counts.
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y = 1x² + 0x + 0
  • Point(1, 1)
  • Slope of the tangent2
a1
Point of tangency1
Figure 1

Warm Up

Straightforward practice. Get the method working first.

5 problems

  1. For (6x + 1)³, what is the inner derivative?

    Answer

  2. What is the most common Chain Rule error?

    • Forgetting to multiply by the inner derivative.
    • Using the wrong power.

  3. For (2x + 7)⁴, the inner derivative?

    Answer

  4. For (x² + 1)³, the inner derivative at x = 2?

    Answer

  5. y = (3x + 1)⁵, so y′ = 15(3x + 1)⁴. What is y′ at x = 0?

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. For (4x − 9)⁷, the inner derivative?

    Answer

  2. Three nested layers means how many factors multiplied?

    Answer

  3. For (x + 5)⁹, the inner derivative?

    Answer

Stretch Yourself

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

4 problems

  1. Put the Chain Rule process in order.

    Write 1 to 4 in the boxes to put these in order.

    • Identify the outer and inner functions.
    • Differentiate the outer, leaving the inside untouched.
    • Differentiate the inner function.
    • Multiply the two results together.

  2. y = (2x)³ expands to 8x³, so y′ = 24x². What is y′ at x = 1?

    Answer

  3. The Inside

    For (5x − 2)³, what is the derivative of the inside?

    Answer

  4. The Chain

    y changes 4 times as fast as u, and u changes 3 times as fast as x. How many times as fast as x does y change?

    Answer