Chapter 2: Differentiation

AP Exam Weight: 25-35% | Multiple Choice: 10-14 questions | Free Response: Major focus in several questions

📚 Table of Contents

1. Definition
2. Basic Rules
3. Advanced Rules
4. Applications
5. Optimization

1. Definition of Derivative 📊

Understanding the Derivative

The derivative measures how quickly a function is changing at any point. Think of it as:

• The instantaneous speedometer in your car
• The slope of a mountain at a specific point
• The rate of population growth at a moment
• The sensitivity of temperature change

Limit Definition

$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

Intuitive Understanding

• The derivative is a special limit
• We're finding the slope of increasingly closer secant lines
• As h gets tiny (approaches 0), secant line becomes tangent line
• Think of zooming in until curve looks like a straight line

Process for Finding Limit Definition

1. Start with difference quotient
2. Substitute and simplify
3. Take limit as h approaches 0
4. Simplify result

Example Walkthrough

Find f'(x) for f(x) = x²

1. Write quotient: $\lim_{h \to 0} \frac{(x+h)^2 - x^2}{h}$
2. Expand: $\lim_{h \to 0} \frac{x^2 + 2xh + h^2 - x^2}{h}$
3. Simplify: $\lim_{h \to 0} \frac{2xh + h^2}{h} = \lim_{h \to 0} (2x + h)$
4. Result: f'(x) = 2x

Alternative Form

$f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x-a}$

When to Use

• Helpful for finding derivatives at specific points
• Useful when function is defined piecewise
• Better for some theoretical proofs
• Equivalent to h-definition

Interpretations

1. Geometric Interpretation

• Slope of tangent line at point (x, f(x))
• Rate of steepness of curve
• Direction of motion along curve
• Visual representation of rate of change

2. Physical Interpretation

• Instantaneous velocity from position function
• Rate of reaction in chemistry
• Rate of growth in biology
• Rate of change in any quantity

3. Practical Applications

• Market rate of change in economics
• Acceleration in physics
• Population growth rates
• Temperature change rates

Notation

Different notations emphasize different aspects of derivatives:

Leibniz Notation: $\frac{d}{dx}[f(x)]$ or $\frac{dy}{dx}$

• Emphasizes ratio of changes
• Helpful for related rates
• Clear variable dependency
• Standard for differential equations

Prime Notation: f'(x)

• Compact and clean
• Good for multiple derivatives (f”, f''')
• Common in theoretical work
• Easy to write quickly

Newton's Notation: $\dot{y}$

• Used mainly in physics
• Represents time derivatives
• Multiple dots for higher derivatives
• Compact for motion problems

Common Misconceptions

1. The Derivative Is Not:

   • Average rate of change
   • Slope between two points
   • Change in y over change in x
   • A secant line slope

2. Common Errors:

   • Forgetting limit process
   • Confusing average and instantaneous rates
   • Misinterpreting negative derivatives
   • Not checking for differentiability

3. Points of Confusion:

   • Corner points
   • Vertical tangent lines
   • Points of discontinuity
   • Infinite derivatives

Practice Strategies

1. Understanding the Definition:

   • Draw secant lines approaching tangent
   • Calculate difference quotients
   • Take limits step by step
   • Verify with graphs

2. Mastering Interpretations:

   • Connect to real-world scenarios
   • Practice slope visualization
   • Link to rate problems
   • Use multiple representations

3. Notation Fluency:

   • Practice converting between notations
   • Use appropriate notation for context
   • Understand meaning of each symbol
   • Connect to physical meanings

2. Basic Rules 📐

Understanding Basic Derivatives

The foundation of calculus rests on these fundamental rules. Think of them as:

• The basic vocabulary of calculus
• Building blocks for more complex derivatives
• Essential patterns to recognize
• Core rules that never change

Power Rule

$\frac{d}{dx}[x^n] = nx^{n-1}$

How It Works

1. Take the exponent
2. Multiply it by the base
3. Reduce exponent by 1
4. Apply to each term separately

Examples

1. $\frac{d}{dx}[x^3] = 3x^2$
2. $\frac{d}{dx}[x^{1/2}] = \frac{1}{2}x^{-1/2}$
3. $\frac{d}{dx}[x^{-2}] = -2x^{-3}$

Common Mistakes

• Forgetting to multiply by power
• Adding instead of subtracting exponent
• Mishandling negative exponents
• Confusion with chain rule cases

Constant Rule

$\frac{d}{dx}[c] = 0$

Understanding Why

• Constants don't change
• Their rate of change is zero
• Graph is horizontal line
• No matter what x does, c stays same

Sum/Difference Rule

$\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)$

Key Concepts

• Derivatives distribute over addition/subtraction
• Each term differentiated separately
• Signs stay the same
• Order doesn't matter

Example Walkthrough

Find $\frac{d}{dx}[3x^4 - 2x^2 + 5x - 7]$

1. Split into terms: $(3x^4) + (-2x^2) + (5x) + (-7)$
2. Apply power rule to each:
   • $3 \cdot 4x^3 = 12x^3$
   • $-2 \cdot 2x^1 = -4x$
   • $5 \cdot 1x^0 = 5$
   • $0$ (constant)
3. Combine: $12x^3 - 4x + 5$

Constant Multiple Rule

$\frac{d}{dx}[cf(x)] = cf'(x)$

Applications

• Pull out coefficients
• Keep constants in place
• Multiply after differentiating
• Useful for simplifying

3. Advanced Rules 🔄

Product Rule

$\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$

Understanding the Rule

• Think "first times derivative of second plus second times derivative of first"
• Like FOIL but for derivatives
• Each function takes turns being differentiated
• Other function stays the same

Common Applications

1. Polynomial Products
   • $(x^2)(x^3)$
   • $(x+1)(x^2-3)$
2. Trig-Polynomial Combinations
   • $x\sin(x)$
   • $x^2\cos(x)$
3. Exponential-Polynomial
   • $xe^x$
   • $x^2\ln(x)$

Example Walkthrough

Find $\frac{d}{dx}[x^2\sin(x)]$

1. Identify f(x) = x² and g(x) = sin(x)
2. Find f'(x) = 2x and g'(x) = cos(x)
3. Apply rule: $(2x)(\sin(x)) + (x^2)(\cos(x))$
4. Result: $2x\sin(x) + x^2\cos(x)$

Quotient Rule

$\frac{d}{dx}[\frac{f(x)}{g(x)}] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$

Memory Device

"Low d-High minus High d-Low over square of below"

Key Points

1. Denominator is always squared
2. Numerator follows specific pattern
3. Order matters in subtraction
4. Check for domain restrictions

Chain Rule

$\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$

Visual Understanding

• Think of peeling layers of an onion
• Work from outside in
• Multiply results together
• Keep track of inside function

Common Patterns

1. Power Functions
   • $(x^2 + 1)^3$
   • $\sqrt{x^2 + 1}$
2. Exponentials
   • $e^{x^2}$
   • $2^{sin(x)}$
3. Trigonometric
   • $\sin(x^2)$
   • $\tan(e^x)$

4. Applications 🎯

Related Rates

Understanding how different changing quantities relate to each other.

Problem-Solving Process

1. Draw a Diagram

   • Label variables
   • Show relationships
   • Mark known values
   • Indicate what's changing

2. Write an Equation

   • Connect variables
   • Use geometric formulas
   • Consider relationships
   • Keep variables as functions of time

3. Differentiate

   • Use chain rule
   • Remember implicit differentiation
   • Keep track of units
   • Watch for constant terms

4. Solve

   • Substitute known values
   • Solve for unknown rate
   • Check units
   • Verify answer makes sense

Common Scenarios

1. Geometric

   • Growing circles
   • Sliding ladders
   • Changing triangles
   • Expanding volumes

2. Real-World

   • Water filling tanks
   • Shadow lengths
   • Distance problems
   • Cost relationships

Linear Approximation

$f(x) \approx f(a) + f'(a)(x-a)$

Understanding the Concept

• Tangent line approximation
• Best near x = a
• First-order approximation
• Local linear behavior

Applications

1. Estimation

   • Quick calculations
   • Error bounds
   • Near-point values
   • Complex functions

2. Error Analysis

   • Maximum error
   • Interval of accuracy
   • Improvement methods
   • Tolerance levels

5. Optimization 📈

Understanding Optimization

Finding maximum or minimum values in real-world situations.

Finding Extrema

Process

1. Identify Variables

   • What changes
   • What's constrained
   • What to optimize
   • Relationship between variables

2. Write Equation

   • Express goal in terms of one variable
   • Use constraints
   • Simplify if possible
   • Check units

3. Find Critical Points

   • Set derivative = 0
   • Find undefined points
   • Check endpoints
   • List all candidates

4. Test Points

   • First derivative test
   • Second derivative test
   • Endpoint comparison
   • Verify answer makes sense

Testing Methods

First Derivative Test

• f'(x) changes - to + : Local minimum
• f'(x) changes + to - : Local maximum
• Check behavior around critical points
• Consider domain restrictions

Second Derivative Test

• f”(x) > 0 : Local minimum
• f”(x) < 0 : Local maximum
• f”(x) = 0 : Inconclusive
• More efficient but not always applicable

Common Applications

1. Geometric

   • Maximum area
   • Minimum perimeter
   • Optimal volume
   • Best dimensions

2. Business

   • Maximum profit
   • Minimum cost
   • Optimal production
   • Best pricing

3. Physics

   • Minimal time
   • Maximum distance
   • Optimal angle
   • Energy efficiency

📝 AP-Style Examples

Example 1: Chain Rule

Find $\frac{d}{dx}[\sin(e^{2x})]$

Solution:

1. Outer function: sin
2. Inner function: $e^{2x}$
3. $f'(x) = \cos(e^{2x}) \cdot e^{2x} \cdot 2$
4. $f'(x) = 2e^{2x}\cos(e^{2x})$

Example 2: Related Rates

A spherical balloon expands at 2 cm³/s. How fast is the radius increasing when r = 5 cm?

Solution:

1. Volume formula: V = 4/3πr³
2. Differentiate: dV/dt = 4πr²·dr/dt
3. Substitute: 2 = 4π(5)²·dr/dt
4. Solve for dr/dt

💡 Success Strategies

1. Choosing the Right Rule

• Multiple terms → Sum/Difference Rule
• Multiplication → Product Rule
• Division → Quotient Rule
• Composition → Chain Rule

2. Common Mistakes

• Forgetting chain rule
• Product vs. power rule confusion
• Sign errors in quotient rule
• Missing terms in product rule

3. Calculator Tips

• Check with numerical derivative
• Graph to verify reasonableness
• Use for complex calculations
• Verify critical points

🔍 AP Exam Focus

Free Response Tips

1. Show all work

   • Rule identification
   • Step-by-step process
   • Final simplification

2. Common Questions

   • Related rates
   • Optimization
   • Chain rule applications
   • Motion problems

Multiple Choice Strategy

1. Consider

   • Basic vs. advanced rules
   • Multiple approaches
   • Common derivatives

2. Check

   • Units
   • Sign
   • Domain restrictions

💡 Pro Tip: Mastering differentiation rules is crucial - they're the foundation for most calculus applications!