Differentiation:
definition and
fundamental properties

Unit 2 overview

Differentiation measures instantaneous change. AP Calculus AB and BC Unit 2 develops the derivative from limits, connects it to tangent slopes and rates, and introduces the fundamental differentiation rules used throughout the course.

Across 10 sequential lessons, you move from average and instantaneous rates to derivative notation, estimation and differentiability. You then build a toolkit of power, linearity, product, quotient, trigonometric, exponential and logarithmic rules. Detailed examples, memory aids and practice in the linked notes help connect a calculation to its meaning.

These ranges follow the current College Board course framework and apply to the multiple-choice section. They are not guaranteed question counts or a fixed weighting for free-response questions.

Key concepts you will master

• Average versus instantaneous change: an average rate uses two distinct inputs; an instantaneous rate is the finite limit of the corresponding difference quotient, when it exists. Interpret slopes and velocity with units.
• The derivative definition: connect the limit to a local rate rather than treating it as an isolated formula.$$\mathrm{f\prime }\left(x\right)=\underset{h\to 0}{lim}\frac{f\left(x+h\right)-f\left(x\right)}{h}$$
• Derivative notation: recognize Lagrange’s prime notation, Leibniz’s $\frac{\mathrm{dy}}{\mathrm{dx}}$ and Newton’s time-derivative dots. Keep a derivative function distinct from its value at one input.
• The power rule: differentiate powers by multiplying by the exponent and lowering that exponent by one, on intervals where the real-valued function and rule apply.
• Linearity: differentiate sums, differences and constant multiples term by term. A constant has derivative zero; a variable factor must not be treated as a constant.
• Core transcendental functions: recall the derivatives of sine, cosine, the natural exponential and natural logarithm. Trigonometric derivative formulas assume radians.
• Product and quotient rules: differentiate combined expressions without confusing the derivative of a product with the product of the derivatives. A quotient rule requires a nonzero original denominator.
• The remaining trigonometric rules: obtain the derivatives of tangent, cotangent, secant and cosecant from sine and cosine, and retain the original domain restrictions.
• Continuity and differentiability: differentiability implies continuity at an interior point, but a continuous function may still fail to have a finite derivative there.
• AP problem-solving: choose a method, preserve notation, explain requested reasoning and check signs, coefficients, domain and units. Memory patterns support understanding rather than replace it.

Learning objectives

By the end of Unit 2, aim to calculate accurately and explain what each result means.

• Calculate a derivative from a limit definition and identify when the required finite limit does not exist.
• Apply the power, constant, sum, difference and constant-multiple rules efficiently.
• Differentiate basic trigonometric, exponential and logarithmic functions with correct notation and domains.
• Use the product and quotient rules step by step, then simplify without losing restrictions.
• Check differentiability and continuity at ordinary points and piecewise joins; recognize why continuity alone is not enough.
• Estimate derivative values from graphs and numerical data, and describe the uncertainty in an estimate.
• Connect a derivative to a contextual rate, tangent slope or velocity, retaining output-per-input units.
• Choose systematic strategies for mixed expressions and explain why the selected rule applies.
• Find and correct common errors through sign, exponent, numerical and structural checks.
• Write the calculations and reasoning requested in AP free-response practice, with clear definitions and conclusions.

Why differentiation matters

A derivative describes how an output responds to a small change in its input. This makes differentiation useful for motion in physics, marginal change in economics and rates in other quantitative models. A formula alone is not the whole interpretation: identify the quantities, point, units and domain.

From an interval rate to a pointwise rate

A difference quotient measures average change over a nonzero input increment:

Taking its finite limit as the increment approaches zero defines the derivative at that input. It is not generally the same as the average rate over a fixed interval.

• Graphical meaning: a derivative value gives a finite tangent slope. Its sign describes local increasing or decreasing behavior under the appropriate differentiability assumptions.
• Modeling: position differentiated with respect to time gives instantaneous velocity. Later applications extend these ideas to acceleration and other rates.
• Exam preparation: Unit 2 is assessed directly and its rules support work in later multiple-choice and free-response problems.
• Reasoning: definitions and differentiability conditions help justify a result, detect invalid shortcuts and explain why a derivative may fail to exist.
• Later calculus: the chain rule and implicit differentiation develop in Unit 3; contextual rate problems appear in Unit 4 and analytical applications such as optimization in Unit 5.
• A lasting toolkit: choosing rules, managing signs and checking a result are habits used in related rates, curve analysis, integration connections and further mathematics.

AP Calculus Unit 2 exam strategy

Six habits for reliable work

1. Know the core formulas. Recall them, then practise selecting the right one from an expression’s structure.
2. Show requested reasoning. Follow the prompt’s requirements; a memorized phrase or unsupported answer does not guarantee credit.
3. Check existence. Look for discontinuities, corners, cusps, vertical tangents or other failures of a finite derivative.
4. Audit signs and order. Recheck trigonometric negatives, power coefficients and quotient subtraction before simplifying.
5. Use precise notation. Distinguish a derivative function, its value at a point and an average rate; include contextual units.
6. Mix the practice. Alternate symbolic, numerical, graphical and verbal questions, and record the reason behind each corrected error.

A flexible five-week study path

This suggested self-study progression preserves a gradual route through the unit. Adjust the pace to your teacher’s course sequence and your current understanding; it is not a required schedule.

1. Week 1 · Meaning and notation · Topics 2.1–2.2

   Compare average and instantaneous change. Build the derivative from a limit and become comfortable reading the different derivative notations.

2. Week 2 · Estimation, existence and basic rules · Topics 2.3–2.6

   Estimate slopes from graphs and tables, test differentiability versus continuity, and develop the power and linearity rules. Explain why a shortcut applies before using it.

3. Week 3 · Core functions, products and quotients · Topics 2.7–2.9

   Memorize and understand the basic trigonometric, exponential and logarithmic derivatives. Practise product and quotient setups, compare alternative rewrites and check domain restrictions.

4. Week 4 · Complete the trigonometric toolkit · Topic 2.10

   Derive and apply the tangent, cotangent, secant and cosecant rules. Review all six trigonometric derivatives, the role of radians and common negative-sign mistakes.

5. Week 5 · Mixed review and exam practice

   Attempt mixed problems and written explanations from the topic notes. Use mistakes as a diagnostic, revisit weak skills and confirm the fundamentals before moving into Unit 3.

What you will find in the topic notes

• Definitions, formulas and conditions: organized reference material for each derivative concept and rule.
• Worked examples: step-by-step reasoning, accurate notation and explanations of why a procedure works.
• Memory aids and efficient methods: recognition patterns and shortcuts used with their required conditions.
• Common mistakes: checks for missing product terms, quotient-order errors, sign errors and incorrect domain assumptions.
• Exam-focused guidance: help presenting calculations and interpreting results in the form a question requests.
• Practice and solutions: opportunities to try a skill before consulting the worked explanation.
• Quick-reference review: concise summaries for revisiting important ideas and connections.
• Readable visual design: properly typeset mathematics, relevant teaching visuals, clear headings and olive/cream callouts.
• Easy navigation: internal connections and topic contents that make the material easier to find, bookmark and revisit.

Use a formula reference to refresh the idea, study an example and then attempt the practice independently. A remembered rule becomes more useful when you can identify its conditions, interpret the answer and explain a common error.

Build your differentiation toolkit

For a first pass, start with Topic 2.1 and follow the cards in sequence. For revision, open the topic matching your weakest skill, work through its examples and test yourself before moving on.

Keep the meaning of a derivative alongside the rules. A reliable calculation, a justified existence check and a clear rate interpretation prepare you for the later applications in both AB and BC.