The Power Rule handles x⁵ but not (3x + 1)⁵. The second is a composition, and compositions need their own rule.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The rule
For f(g(x)) the derivative is f′(g(x)) · g′(x): differentiate the outer function, leave the inside alone, then multiply by the inside derivative.
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. Rates multiply along a chain.
Finding the inside
The inside is whatever sits under a power, inside a root, or inside a trigonometric 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.
Outside then inside
For f(g(x)), the derivative is f′(g(x)) · g′(x). Differentiate the outer function leaving the inner alone, then multiply by the inner derivative. Recognising the composition is the whole difficulty.
Why the inner derivative appears
If the inner function changes three times as fast as x, the composition changes three times as fast as it otherwise would. The rates multiply, which is exactly what the rule records.
In Leibniz notation it looks obvious
dy/dx = dy/du · du/dx, with the du appearing to cancel. That is not a proof, but it is why Leibniz notation is preferred here — the form of the rule is visible in the symbols.
Nesting goes as deep as needed
For three layers, multiply three derivatives. Working from the outside inwards and writing each factor as you go keeps track of them. Missing an inner factor is the standard error and it is caught by checking the layer count.
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 Tomas Differentiate (3x + 1)⁵
Tomas must differentiate this without expanding the bracket.
- Step 1
The outer function is the fifth power and the inner function 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, what is the inner derivative?
2 of 8
y = (3x)². Expanding gives 9x², so y′ = 18x. What is y′ at x = 1?
3 of 8
For (x² + 1)³, what is the inner derivative at x = 2?
4 of 8
y = (3x + 1)⁵, so y′ = 15(3x + 1)⁴. What is y′ at x = 0?
5 of 8
For (4x − 9)⁷, what is the inner derivative?
6 of 8
Three nested layers means how many factors multiplied?
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
For (x + 5)⁹, what is the inner derivative?
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, leaving the inside untouched, then multiply by the inside derivative.
- Rates multiply along a chain, which is why the factors multiply.