The chain rule
Objective. Differentiate a function of a function.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
If y = f(g(x)) then dy/dx = f′(g(x)) × g′(x): differentiate the outside, keep the inside, then multiply by the derivative of the inside.
Worked example
Differentiate y = (3x² + 1)⁵.
- 1Outside: 5(3x² + 1)⁴.
- 2Inside derivative: 6x.
- 3Multiply: 30x(3x² + 1)⁴.
Answer: 30x(3x² + 1)⁴
Concept mastery
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Practice
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Start practisingCommon mistake. Forgetting to multiply by the derivative of the inner function.
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