The Chain Rule

Differentiate the outside, then the inside.

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

What this lesson teaches

The student differentiates composite functions using the Chain Rule.

The Chain Rule gives f′(g(x)) · g′(x).
Differentiate the outside first, leaving the inside untouched, then multiply by the inside derivative.
Rates multiply along a chain, which is why the factors multiply.

Straightforward practice. Get the method working first.

5 problems

For (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, what is the inner derivative?
Answer 2
Why Differentiate 2x + 7.
y = (3x)². Expanding gives 9x², so y′ = 18x. What is y′ at x = 1?
Answer 18
Why 18 × 1.
For (x² + 1)³, what is the inner derivative at x = 2?
Answer 4
Why 2x at x = 2.

The same ideas with more to keep track of.

3 problems

y = (3x + 1)⁵, so y′ = 15(3x + 1)⁴. What is y′ at x = 0?
Answer 15
Why 15 × 1.
For (4x − 9)⁷, what is the inner derivative?
Answer 4
Why Differentiate 4x − 9.
Three nested layers means how many factors multiplied?
Answer 3
Why One per layer.

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

4 problems

Put 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.
For (x + 5)⁹, what is the inner derivative?
Answer 1
Why The derivative of x + 5.
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.