Chain Rule explained step by step
The chain rule differentiates a composite function—a function placed inside another function. If y = f(g(x)), then y′ = f′(g(x))·g′(x). In practical terms, differentiate the outside function while leaving the inside unchanged, then multiply by the derivative of the inside. For (3x + 1)², the outside derivative is 2(3x + 1), and the inside derivative is 3, so the result is 6(3x + 1). The main skill is recognizing the layers before calculating. The most effective way to improve is to study one concept at a time, verify it with a question, and revisit it later. This turns every mistake into a clear decision for your next study session.
How to use this guide
- 1Read one idea at a time
- 2Explain it in your own words
- 3Practice it and review the mistake
Where the chain rule is used
Look for an operation applied to an entire expression.
Powers
(x² + 4)⁵ contains a power outside a polynomial.
Trigonometric functions
sin(2x) requires the derivative of sine and the derivative of 2x.
Exponential and logarithmic functions
e^(x²) and ln(5x − 1) are composite functions.
Multiple layers
For nested compositions, move from the outermost layer inward and multiply each derivative.
A worked example
Differentiate f(x) = (2x³ − 5)⁴ by labeling the layers.
Outer derivative
The derivative of u⁴ is 4u³, giving 4(2x³ − 5)³.
Inner derivative
The derivative of 2x³ − 5 is 6x².
Multiply
f′(x) = 4(2x³ − 5)³ · 6x².
Simplify
f′(x) = 24x²(2x³ − 5)³.
Learn the structure, not a slogan
Tutorlify asks you to identify each function layer, provides a gradual hint, and changes the example when you need another representation.
Practice the chain ruleLabel outer and inner
Differentiate each layer
Multiply the derivatives
Transfer to a new example
Worked examples
Powers
(x² + 4)⁵ contains a power outside a polynomial.
- Identify the central idea.
- Explain it in your own words.
- Check how it applies to a concrete case.
The answer should connect where the chain rule is used with a clear explanation and concrete evidence.
Inner derivative
The derivative of 2x³ − 5 is 6x².
- Identify the central idea.
- Explain it in your own words.
- Check how it applies to a concrete case.
The answer should connect a worked example with a clear explanation and concrete evidence.
Practice exercises
Practice 1
Explain “Outer function” without checking the guide, then create your own example.
Show hint
Start with a short definition, add one important relationship, and finish with an application.
Practice 2
Explain “Inner function” without checking the guide, then create your own example.
Show hint
Start with a short definition, add one important relationship, and finish with an application.
Practice 3
Explain “Layer-by-layer reasoning” without checking the guide, then create your own example.
Show hint
Start with a short definition, add one important relationship, and finish with an application.
Frequently asked questions
01Do I always multiply when using the chain rule?+
Yes. Multiply the derivative of the outside by the derivative of the inside.
02Can chain rule appear with product rule?+
Yes. Apply the product rule to the factors and use chain rule inside any composite factor.
03What is the most common mistake?+
Differentiating the outer function but forgetting the derivative of the inner expression.
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