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 why reflecting a graph in y = x turns a slope m into 1/m at the matching point. (Guide: “The idea”, Figure 1; Practice Q3)
- I can explain the difference between f⁻¹(x) and 1/f(x). (Guide: “A note on notation”; Practice Q2)
- I can explain why a horizontal tangent on f gives a vertical tangent on its inverse. (Guide: “When the inverse is not differentiable”; Practice Q4, Q7(d))
Skills
- I can derive g′(x) = 1/f′(g(x)) by differentiating f(g(x)) = x with the chain rule, or by implicit differentiation. (Guide: “Deriving the rule with the chain rule”)
- I can find the matching input a with f(a) = b before using f′(a). (Guide: Worked example 1; Practice Q1, Q7(b))
- I can read an inverse’s values and slopes from a table of f and f′. (Guide: Worked example 2; Practice Q2)
- I can write the tangent line to the graph of an inverse at a given point. (Guide: Worked examples 1 and 2; Practice Q6(c))
- I can combine the inverse rule with the chain rule or product rule. (Guide: Worked example 2(d); Practice Q6(d))
- I can give units for the derivative of an inverse in a context and interpret it. (Practice Q5)
Reasoning
- I can show that a function is one-to-one by showing its derivative keeps one sign. (Guide: Worked example 1; Practice Q5(a), Q6(a))
- I can confirm an answer by checking the rule on a known inverse pair, such as eˣ and ln x, or with a nearby value. (Guide: “Checking the rule”; Practice Q7(c))
- I can spot the “1/f′(b)” error in someone else’s working and correct it. (Practice Q7(a))
All ticked? Move on to Topic 3.4, Differentiating Inverse Trigonometric Functions, or return to the Calculus AB roadmap or the Calculus BC roadmap.