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 find the derivative of x³ from the limit definition and explain how the pattern n·xⁿ⁻¹ arises. (Guide: “Building the pattern from the definition”)
- I can use the limit definition to confirm the power rule for a negative power such as 1/x². (Guide: “Testing the pattern on 1/x and √x”; Practice Q5)
- I can state the power rule, including the condition that the exponent is a constant. (Guide: “The power rule”; Revision notes)
- I can explain why the power rule does not apply to 2ˣ or to a constant such as π³. (Guide: “Common misconceptions”)
Skills
- I can rewrite reciprocals and roots as powers of x, such as 1/x⁵ = x⁻⁵ and ∜(x³) = x^(3/4). (Guide: “Rewrite before you differentiate”; Practice Q1, Q2)
- I can lower a negative or fractional exponent by 1 without sign errors. (Guide; Practice Q1, Q6(b))
- I can combine products and quotients of powers into a single power before differentiating. (Guide: Worked example 1; Practice Q6)
- I can work backwards from a derivative to the power function it came from. (Practice Q3)
- I can find the slope and equation of a tangent line to y = xʳ at a given point. (Guide: Worked example 2; Practice Q4, Q7(b))
- I can find the x-values where a tangent to a power function has a given slope, including negative solutions. (Practice Q4, Q7(c))
Reasoning
- I can explain why the derivative of a product of powers is not the product of their derivatives, using a numerical counterexample. (Practice Q6(d))
- I can identify where the power rule gives an undefined derivative and use the definition to decide between a vertical tangent and a cusp. (Guide: “When the power rule gives ‘undefined’”; Practice Q7(d))
- I can explain why the domain of f′ can be smaller than the domain of f. (Guide: “When the power rule gives ‘undefined’”)
All ticked? Move on to Topic 2.6, Derivative Rules: Constant, Sum, Difference, and Constant Multiple, or return to the Calculus AB roadmap or the Calculus BC roadmap.