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Calculus BC · AP · Differentiation — Definition and Fundamental Properties

What this topic is

Students apply tools and theorems to new situations: Applying the Power Rule. Watch for correct setup first; arithmetic can be cleaned up after the plan is right.

This topic is assessed on the AP Exam.

Why it matters

• Derivative fluency here is reused in related rates, optimization, and motion problems later.
• The derivative definition and basic rules are core AP Calculus skills used all year.
• Students must connect average rate, instantaneous rate, and slope of the tangent line.
• Power, product, and quotient rules appear constantly on multiple-choice.
• Correct units on rates (e.g., miles per hour) earn communication points on FRQs.

How you can support

• Ask them to state what the derivative means in the problem’s units before simplifying.
• Ask: “Is this an average rate over an interval or an instantaneous rate at a point?”
• Have them write the limit definition once, then use a shortcut rule on a second item.
• Celebrate a correct derivative expression before simplifying fully.
• Ask what the derivative means in the story (speed, growth, slope).
• If they use a calculator derivative, still ask for the rule that justifies it.

What not to do

• Do not skip naming which differentiation rule justifies each step.
• Do not confuse f′(a) with the average rate on [a, b].
• Do not drop units on contextual rate answers.
• Do not apply the power rule to products without the product rule.
• Do not treat “derivative of a constant” as optional—zeros matter.