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 that f″ is the derivative of f′, and that f‴, f⁽⁴⁾ and so on follow the same way. (Guide: “The idea”; Practice Q1, Q3)
- I can read and write f″(x), y″, d²y/dx², f⁽ⁿ⁾(x) and dⁿy/dxⁿ, and say why d²y/dx² is not (dy/dx)². (Guide: “Notation you must recognise”; Practice Q2, Q4)
- I can explain why f″(a) may fail to exist even when f′(a) exists, with an example. (Guide: the x^(4/3) example; Practice Q6)
Skills
- I can find a second derivative that needs the product, quotient or chain rule, simplifying f′ first. (Guide: Worked example 1; Practice Q3, Q7(a))
- I can find the nth derivative of a polynomial, of e^(kx) and of sin(kx) or cos(kx) using a pattern. (Guide: “Patterns” and Figure 1; Practice Q2, Q4)
- I can find d²y/dx² for an implicit curve, substituting dy/dx and using the curve’s equation. (Guide: Worked example 2; Practice Q5)
- I can evaluate a higher derivative at a point from its formula. (Practice Q1, Q4, Q7(d))
Reasoning
- I can justify, with one-sided limits or a corner in f′, that a second derivative does not exist at a point. (Practice Q6(d))
- I can suggest a formula for g⁽ⁿ⁾(x) from the first few derivatives and check that it carries on to n + 1. (Practice Q7)
- I can explain why y cannot be treated as a constant in an implicit second derivative. (Guide: “Common misconceptions”; Practice Q5(c))
All ticked? Move on to Topic 4.1: Interpreting the Meaning of the Derivative in Context, or return to the Calculus AB roadmap or the Calculus BC roadmap.