(69) AP Calculus AB and BC Unit 2 Review [Differentiation: Definition and Basic Derivative Rules]

Overview of Unit 2

• Focuses on derivatives, building on concepts from Unit 1 (limits and continuity).

• Key foundational rules for Units 3-5 will be introduced.

• Emphasis on memorization of derivative formulas for quicker exam responses.

Average vs. Instantaneous Rate of Change

• Average Rate of Change: Calculated using the difference quotient.

  • Can be expressed as:

    • ( \frac{f(a + h) - f(a)}{h} )

    • or ( \frac{f(x) - f(a)}{x - a} )

• Instantaneous Rate of Change: Derivative at a specific point, transitioning from average rate of change.

  • Achieved by narrowing the interval between two points until they are infinitesimally close.

  • Graphically represented by the slope of the tangent line at that point.

Difference Quotient

• Example to calculate average rate of change between two points

  • Given points (1, 3) and (9, 6):

    • Using the formula ( \frac{f(x) - f(a)}{x - a} ): ( \frac{6 - 3}{9 - 1} = \frac{3}{8} )

  • Interpretation: For every horizontal unit, vertical movement is ( \frac{3}{8} ) units.

Definition of the Derivative

• Found by applying limits to the difference quotient formulas:

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

• Both forms show how to find the instantaneous rate of change.

Notation of the Derivative

• Common forms: ( f'(x) ), ( y' ), ( \frac{dy}{dx} ).

• All forms represent the first derivative and are interchangeable.

Example of Finding Derivative

• For ( f(x) = x^2 + 1 ) at ( a = 3 ):

  • Derive using the definition: ( f'(3) = \lim_{x \to 3} \frac{f(x) - f(3)}{x - 3} )

  • Calculates to give an instantaneous rate of change of 6.

Estimating Derivatives

• Estimation from graphs or tables of values:

  • Example: Estimating derivative at ( x=3 ) using values around it.

  • Use difference quotient with points on either side:

    • Example Calculation: ( f(4) = 5.2 ) and ( f(2) = 3.5 )

  • Yielded a derivative estimate of -0.85.

Differentiability and Continuity

1. If a function is differentiable at a point, then it is continuous at that point.

2. A continuous function is not necessarily differentiable.

3. If a function is discontinuous at a point, it is not differentiable at that point.

4. Smoothness is necessary for differentiability — non-smooth points (e.g., cusps) imply non-differentiability.

Piecewise Functions & Differentiability

• Examining piecewise functions to determine differentiability.

  • Sketch the function accurately according to given conditions.

  • Identify points of continuity and smoothness.

Power Rule

• The simplest rule for derivatives:

  • For traditional power functions: ( rac{d}{dx} x^n = nx^{n-1} )

  • Example: For ( 3x^4 ): ( f'(x) = 12x^3 )

  • Applies similarly for negative exponents.

Derivative of Constants and Polynomials

• Derivative of a Constant: Always zero.

• Derivative of a Polynomial: Derivative of each term computed separately.

  • Example: ( f(x) = x^3 + 3x^2 + 1 ) yields ( 3x^2 + 6x ).

Derivatives of Trigonometric Functions

• Important derivatives to memorize for the AP exam:

  • ( \frac{d}{dx} \sin(x) = \cos(x) )

  • ( \frac{d}{dx} \cos(x) = -\sin(x) )

  • Continue the pattern of derivatives of sine and cosine with respect to fluctuations in signs.

• Exponential function derivative: ( \frac{d}{dx} e^x = e^x )

• Logarithmic function derivative: ( \frac{d}{dx} \ln(x) = \frac{1}{x} )

Product Rule

• Formula to find the derivative of a product of functions:

  • ( (f imes g)' = f'g + fg' )

  • Example: For ( 6x ext{sin}(x) ): ( f' = 6 ) and ( g' = \cos(x) )

  • Result: ( 6 ext{sin}(x) + 6x ext{cos}(x) )

Quotient Rule

• Used when differentiating a rational function:

  • Formula: ( \frac{d}{dx} \left( \frac{f}{g} \right) = \frac{f'g - fg'}{g^2} )

  • Example: For ( \frac{x^2}{\ln(x)} ): Leading to the simplification of ( \frac{2x\ln(x)-x}{\ln(x)^2} )

Derivatives of Other Trigonometric Functions

• Extend differentiation rules for tangent, cotangent, secant, and cosecant.

• Can often simplify complex derivatives using base definitions of functions.

Summary of Key Concepts

• Understanding average vs. instantaneous rates of change is crucial.

• Know definitions of derivatives and can apply limits to difference quotients.

• Ensure familiarity with derivative notations and formulas, particularly for key functions and rules (product and quotient).

• Ability to relate smoothness and continuity to differentiability is vital for problem-solving.