Work through the list without notes or a calculator. If you cannot do a statement, follow its link, then come back. Guide = study guide; Practice = practice questions. This topic is shared by Calculus AB and Calculus BC.
Understanding
- I can explain, with an example like (3x + 1)², why the power rule alone gives the wrong derivative of a composite. (Guide: “Why a new rule is needed”)
- I can split a composite into its outer and inner functions, including hidden ones such as tan³x and 1/(x² + 1). (Guide: “Outer and inner functions”; Practice Q5(a))
- I can explain why rates multiply along a chain, using units such as (points/°C) × (°C/hour). (Guide: Figure 1; Practice Q4)
Skills
- I can differentiate a power, root, trig, exponential or log of an inner function, multiplying by the derivative of the inside. (Guide: Worked example 1; Practice Q1, Q2)
- I can find the derivative of f(g(x)) at a point from a table, finding g(a) before I look up f′. (Guide: Worked example 2; Practice Q3, Q6(a), Q6(b))
- I can apply the chain rule several times to a function with three or more layers. (Guide: “More than two layers”; Practice Q5)
- I can use the chain rule inside the product rule and simplify the result. (Guide: Worked example 3; Practice Q6(c))
- I can use a chain-rule derivative to write a tangent line and to find where a tangent is horizontal. (Guide: Worked examples 1 and 3; Practice Q7)
Reasoning
- I can decide which rule is the outer structure by asking which operation is done last. (Guide: Worked example 3, “Order of operations”)
- I can find and correct a missing or wrong factor in someone else’s chain-rule answer. (Guide: “Common misconceptions”; Practice Q5(b))
- I can tell sin²x apart from sin(x²), and a power of a function such as [f(x)]³ apart from a function of a power such as g(x²), and differentiate each correctly. (Revision notes: “Mistakes to avoid”; Practice Q6(a), Q6(b))
All ticked? Move on to Topic 3.2, Implicit Differentiation, or return to the Calculus AB roadmap or the Calculus BC roadmap.