AP Calculus Unit 2: Differentiation — Definition & Fundamental Properties

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AP Calculus Unit 2 introduces differentiation and the fundamental properties of derivatives. Learn how derivatives are defined, interpreted, and calculated using limits, derivative notation, and essential differentiation rules. Explore the complete Unit 2 Mega Smart Notes and follow the accompanying video lessons to master each concept step by step.

2.1 Average & Instantaneous Rates of Change (Mega Smart Notes)

Learn how to calculate and interpret average and instantaneous rates of change. Explore average rate of change as the slope of a secant line, work with functions, tables, graphs, and piecewise functions, and see how shrinking an interval transforms the secant line into a tangent line. Finally, connect instantaneous rate of change to the derivative and learn how to recognize derivative expressions written as limits.

▶ Video 1: Average Rate of Change Explained: Formula, Secant Lines & Examples
▶Video 2: Average Rate of Change from Tables & Graphs Step-by-Step Examples
▶Video 3: Average Rate of Change: How to Find an Unknown Endpoint
▶Video 4: Instantaneous Rate of Change: From Secant Line to Derivative
▶Video 5: Recognizing the Derivative from a Limit: The Patterns You MUST Know

2.2  Defining the Derivative and Using Derivative Notation  (Mega Smart Notes)

Learn how to define derivatives using limits, use different forms of derivative notation, and connect derivatives to instantaneous rates of change and tangent-line slopes. Explore how to calculate derivatives from first principles, write tangent-line equations, interpret derivative information from graphs, and solve problems involving slopes, points, and real-world rates of change.

▶ Video 1: Derivative Definition & Notation: The Two Limit Forms You Must Know
▶Video 2: Finding Derivatives Using the Limit Definition | 4 Step-by-Step Examples
▶Video 3: Equation of a Tangent Line: Find It Using the Derivative | Step-by-Step
▶Video 4: Tangent Lines from Graphs: How to Find  f'(a) and the Tangent Equation
▶Video 5: Find Where the Tangent Has a Given Slope | Derivative from First Principles
▶Video 6: Advanced Tangent Line Problems: External Points, Parallel Lines & Parameters
▶Video 7: Interpreting Derivatives in Context: Meaning, Units & Rates of Change

2.3 Estimating Derivatives of a Function at a Point (Mega Smart Notes)

Learn how to estimate derivatives using tables, graphs, and numerical approximations. Explore left-hand, right-hand, and central difference quotients, estimate instantaneous rates of change, and interpret tangent-line slopes from graphical information. Practice AP Calculus-style problems and learn how to avoid common mistakes when estimating derivatives.

▶ Video 1: Estimating Derivatives from a Table. Left, Right & Central Difference Quotients
▶Video 2: Estimating Derivatives from a Graph.  Find f'(a) from the Tangent Slope
▶Video 3: Estimating Derivatives. Common Mistakes You MUST Avoid

2.4 Connecting Differentiability and Continuity  (Mega Smart Notes)

Learn the essential connection between differentiability and continuity and understand why differentiability guarantees continuity, but continuity does not always guarantee differentiability. Explore piecewise functions, test continuity and matching derivatives at boundary points, and identify where functions fail to be differentiable because of discontinuities, corners, cusps, or vertical tangents. Finally, master the key logical implications and counterexamples connecting derivatives, continuity, and limits.

▶ Video 1: Differentiability vs Continuity: What's the Connection? | Calculus Explained
▶Video 2: Piecewise Functions: How to Make a Function Differentiable | Find a and b 
▶Video 3: Where Is a Function NOT Differentiable? | Corners, Cusps & Vertical Tangents
▶Video 4: Differentiable, Continuous or Limit Exists? The Logic You MUST Know 

2.5 Applying the Power Rule in Differentiation  (Mega Smart Notes)

Learn how to differentiate functions quickly using the Power Rule and combine it with the Constant, Constant Multiple, and Sum/Difference Rules. Work with positive, negative, and fractional exponents, including fractions and radicals, evaluate derivatives at specific points, and learn how to recognize limits as derivatives so you can apply differentiation rules instead of unnecessary algebra.

▶ Video 1: The Power Rule: Differentiate Powers, Fractions & Radicals | Calculus
▶Video 2: Power Rule with Constants, Sums & Polynomials
▶Video 3: Find the Derivative at a Point | Differentiate First, Then Evaluate
▶Video 4: Recognizing a Derivative from a Limit    

2.6 Derivative Rules — Constant, Sum, Difference, and Constant Multiple (Mega Smart Notes)

Learn how to apply the Constant, Constant Multiple, Sum, and Difference Rules to differentiate functions efficiently. Practice evaluating derivatives at specific points, using given derivative values, solving equations involving derivatives, and applying differentiation rules to information from graphs.

▶ Video 1: 4 Derivative Rules You MUST Know | Constant, Sum, Difference & Multiple
▶Video 2: How to Evaluate Derivatives at a Point | Differentiate First, Then Substitute
▶Video 3: Given f'(a)? How to Use Derivative Information | Step-by-Step
▶Video 4: Finding Derivatives from Graphs | Use Slope + Derivative Rules

2.7 Derivatives of cos x, sin x, eˣ, and ln x (Mega Smart Notes)

Learn the essential derivatives of sin x, cos x, eˣ , and ln x and combine them with familiar differentiation rules. Practice evaluating derivatives at specific points and recognizing derivative definitions hidden inside limits, including exponential-function limits. 

▶ Video 1:  Derivatives of sin x, cos x, eˣ and ln x
▶Video 2: Evaluate Derivatives at a Point Trig, Exponential & Log Functions
▶Video 3:  Limits That Are Actually Derivatives | Recognize the Pattern!

2.8 The Product Rule (Mega Smart Notes)

Learn how to apply the Product Rule to differentiate products of functions involving powers, trigonometric, exponential, and logarithmic functions. Practice evaluating derivatives at specific points, using derivative information from tables and graphs, and solving more advanced problems involving unknown parameters and products of three functions.

▶ Video 1: The Product Rule Explained  
▶Video 2: Product Rule at a Point
▶Video 3: Product Rule from Tables & Graphs | How to Find Derivatives Without a Formula
▶Video 4: Advanced Product Rule | Find Parameters & Differentiate 3 Functions

2.9 The Quotient Rule (Mega Smart Notes)

Learn how to apply the Quotient Rule to differentiate functions written as ratios. Practice using the rule with algebraic, trigonometric, exponential, and logarithmic functions, finding derivatives and tangent lines at specific points, solving for points with a given tangent slope, and working with derivative information from tables and more advanced function combinations.

▶ Video 1: The Quotient Rule Explained | Formula & 4 Step-by-Step Examples
▶Video 2: Quotient Rule Practice | 3 Step-by-Step Derivative Examples
▶Video 3: Quotient Rule Applications | Tangent Lines, Slopes & Derivatives
▶Video 4: Find Points with a Given Tangent Slope | Quotient Rule Calculus
▶Video 5: Advanced Quotient Rule Problems | Tables, Trig, Logs & Square Roots

2.10 Derivatives of tan x, cot x, sec x, and csc x (Mega Smart Notes)

Learn the essential derivatives of tan x, cot x, sec x, and csc x and understand how they can be derived using the Quotient Rule. Apply these formulas with trigonometric identities, Product and Quotient Rules, simplify expressions before differentiating, and recognize derivative definitions hidden inside limits.

▶ Video 1: Derivatives of tan, cot, sec & csc | 4 Trig Rules Explained & Proven
▶Video 2: Trig Derivative Practice | Simplify BEFORE You Differentiate!
▶Video 3: Product Rule with Trig Functions | tan, sec & csc Derivatives Explained
▶Video 4: Quotient Rule with Trig Functions | Derivatives & Hidden Limits Explained

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