Cogito
AP Calculus AB · Chapter 3 · Lesson 2
The Chain Rule
Rates multiply along a chain.
12 problems · about 22 minutes · FUN-3.C
What this lesson teaches
The student differentiates composite functions, including nested compositions.
- The Chain Rule gives f′(g(x)) · g′(x).
- Differentiate the outside first, then multiply by the inside derivative.
- Rates multiply along a chain, which is why the factors multiply.
Warm Up
Straightforward practice. Get the method working first.
5 problemsFor (6x + 1)³, what is the inner derivative?
Answer 6
Why 6.
What is the most common Chain Rule error?
Answer Forgetting to multiply by the inner derivative.
Why Dropping the inner derivative.
For (2x + 7)⁴, the inner derivative?
Answer 2
Why Differentiate 2x + 7.
For (x² + 1)³, the inner derivative at x = 2?
Answer 4
Why 2x at x = 2.
y = (3x + 1)⁵, so y′ = 15(3x + 1)⁴. What is y′ at x = 0?
Answer 15
Why 15 × 1.
Build It Up
The same ideas with more to keep track of.
3 problemsFor (4x − 9)⁷, the inner derivative?
Answer 4
Why Differentiate 4x − 9.
Three nested layers means how many factors multiplied?
Answer 3
Why One per layer.
For (x + 5)⁹, the inner derivative?
Answer 1
Why The derivative of x + 5.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsPut the Chain Rule process in order.
Answer 1. Identify the outer and inner functions. 2. Differentiate the outer, leaving the inside untouched. 3. Differentiate the inner function. 4. Multiply the two results together.
Why Identifying the layers comes before differentiating anything.
y = (2x)³ expands to 8x³, so y′ = 24x². What is y′ at x = 1?
Answer 24
Why 24 × 1.
The Inside: For (5x − 2)³, what is the derivative of the inside?
Answer 5
Why 5.
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 12
Why 12.