For f(g(x)) the derivative is f′(g(x)) · g′(x). Differentiate the outer function, leave the inside alone, then multiply by the inner derivative.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Why it multiplies
If y changes 3 times as fast as u, and u changes 2 times as fast as x, then y changes 6 times as fast as x.
Finding the inside
The inside is whatever sits under a power, inside a root, or inside a trigonometric, exponential or logarithmic function.
The commonest error
Forgetting the inner derivative. It is invisible when the inside is plain x, because that derivative is 1.
Deeper nesting
Three layers means three factors. Work from the outside in, multiplying as you go.
Combined with other rules
A product of composites needs both rules. Apply the product rule first, then the chain rule inside each piece.
Outside, then inside
The derivative of f(g(x)) is f′(g(x))·g′(x). Differentiate the outer function leaving the inner intact, then multiply by the inner derivative. Recognising the composition is most of the difficulty.
Rates multiply along a chain
If the inner function changes three times as fast as x, the composition changes three times as fast as it otherwise would. The rule records that multiplication of rates, which is why dy/dx = dy/du · du/dx.
Layers multiply
Three nested functions give three factors. Working from the outside inwards and writing each factor as it appears prevents the standard error, which is losing an innermost derivative.
It underlies almost everything later
Implicit differentiation, related rates and integration by substitution are all the chain rule in different clothing. Fluency here pays off through the rest of the course more than any other single rule.
Step 2: Try It Yourself
Tap and try it out.
- Point(1, 1)
- Slope of the tangent2
Step 3: Watch an Example
One step at a time.
Watch Marcus Differentiate a Composite
Marcus differentiates y = (3x + 1)⁵ without expanding.
- Step 1
The outer function is the fifth power and the inner is 3x + 1.
Step 4: Your Turn
Practice makes it stick.
The Inside
Problem 1 of 2
For (5x − 2)³, what is the derivative of the inside?
The Chain
Problem 2 of 2
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?
Outside Then Inside
1 of 8
For (2x + 7)⁴, the inner derivative?
2 of 8
For (x² + 1)³, the inner derivative at x = 2?
3 of 8
y = (3x + 1)⁵, so y′ = 15(3x + 1)⁴. What is y′ at x = 0?
4 of 8
For (4x − 9)⁷, the inner derivative?
5 of 8
Three nested layers means how many factors multiplied?
6 of 8
For (x + 5)⁹, the inner derivative?
7 of 8
Put the Chain Rule process in order.
- 1Differentiate the outer, leaving the inside untouched.
- 2Differentiate the inner function.
- 3Multiply the two results together.
- 4Identify the outer and inner functions.
8 of 8
y = (2x)³ expands to 8x³, so y′ = 24x². What is y′ at x = 1?
Step 5: Quick Check
Show what you know.
Question 1 of 2
For (6x + 1)³, what is the inner derivative?
Question 2 of 2
What is the most common Chain Rule error?
What You Learned
- 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.