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 state the derivatives of sin x, cos x, eˣ and ln x, with the correct signs and the condition x > 0 for ln x. (Guide: “The four new rules”; Practice Q1, Q3)
- I can explain, using the limit definition and the Topic 1.8 trig limits, why d/dx sin x = cos x. (Guide: “Why the trig rules are true”)
- I can use a graph to show that the slopes of sin x match the heights of cos x. (Guide: Figure 1)
- I can explain why eˣ is its own derivative and why the slope of ln x is 1/x. (Guide: “Why eˣ is its own derivative”, Figure 2; Practice Q7(d))
- I can explain why the trig rules need radians. (Guide: “Why radians matter”)
Skills
- I can differentiate a sum of constant multiples of sin x, cos x, eˣ, ln x and powers of x, giving constants a derivative of 0. (Guide: “Rewrite before you differentiate”; Practice Q1, Q3)
- I can find the equation of a tangent line to a graph built from these functions. (Guide: Worked example 1; Practice Q5(b), Q7)
- I can find a point where the slope has a given value, or show that no such point exists. (Guide: Worked example 2; Practice Q4)
- I can evaluate a limit by recognising it as f′(a), naming f and a. (Guide: Worked example 3; Practice Q2, Q6)
Reasoning
- I can tell which expressions this topic covers and which need the chain rule, such as sin(3x). (Guide: “Rewrite before you differentiate”)
- I can use the intermediate value theorem on f′ to justify that a horizontal tangent exists, stating continuity. (Practice Q5(c))
- I can catch the common slips: the sign of d/dx cos x, the power rule on eˣ, and differentiating constants such as e². (Guide: “Common misconceptions”; Revision notes)
All ticked? Move on to Topic 2.8, the product rule, or return to the Calculus AB roadmap or the Calculus BC roadmap.