Differentiation: Definition and Fundamental Properties

AP Calculus AB: Unit 2 - Differentiation: Fundamental Properties & DefinitionsIntroduction: What is a Derivative? At its core, the derivative has two fundamental interpretations:Geometrically: The derivative of a function f(x) at a point x=c is the slope of the tangent line to the graph of f(x) at that point.Physically: The derivative represents the instantaneous rate of change of one quantity with respect to another. (e.g., velocity is the derivative of position).1. Defining the Derivative at a PointThe derivative is formally defined using limits. It's the limit of the slope of a secant line as the two points on the line get infinitely close to each other.Formal Definitions (MEMORIZE BOTH)a) The Limit Definition of the Derivative at a Point c:The derivative of a function f at a point c, denoted f'(c), is:f(c): The y-value at the point of tangency.f(c+h): The y-value of a nearby point.h: The small change in x between the two points.f(c+h) - f(c): The change in y (rise).(f(c+h) - f(c)) / h: The slope of the secant line (average rate of change).b) The Alternate Form:This form is often useful for theoretical problems or specific limit calculations.Example: Find the derivative of at using the limit definition.Using the first form with : Conclusion: The slope of the tangent line to at is 6.Notation for the Derivative at a Point:2. The Derivative as a FunctionInstead of finding the slope at a single point c, we can create a new function, f'(x), that gives the slope at any point x where the derivative exists. We do this by simply replacing c with x in the definition.Example: Find the derivative function f'(x) for .So, the derivative of is . Now we can find the slope at any point easily: at , the slope is . At , the slope is .3. Connection Between Differentiability and ContinuityThis is a critical theoretical concept for the AP Exam.Theorem: Differentiability implies ContinuityIf a function f is differentiable at a point x=c, then f MUST be continuous at x=c.Why? For a tangent line to exist (i.e., a defined slope), the function must be "connected" at that point. You can't have a defined slope at a hole or a jump.Converse is FALSE: Continuity does NOT imply DifferentiabilityA function can be continuous at a point but fail to be differentiable there.Three ways a function can fail to be differentiable at a point x=c:Corner: The slopes from the left and right are different.Example: at . The slope from the left is -1, and the slope from the right is +1. Since they don't match, the derivative at does not exist.Cusp: Similar to a corner, but the slopes from each side approach and .Example: at .Vertical Tangent Line: The slope of the tangent line is infinite (undefined).Example: at . The tangent line is the y-axis, which has an undefined slope.Discontinuity: (Jump, Hole, Asymptote) If it's not continuous, it can't be differentiable.4. Basic Differentiation Rules (The Shortcuts!)These rules allow us to bypass the limit definition for most functions.The Constant Rule:(The slope of a horizontal line is zero).The Power Rule (Most Important Rule):(Bring the power down, then subtract one from the power).Examples:The Constant Multiple Rule:(Constants "ride along" with the derivative).Example: The Sum and Difference Rule:(You can take the derivative of each term separately).Example (Polynomials):5. Product and Quotient RulesUsed when differentiating functions that are multiplied or divided.The Product Rule:Mnemonic: "First times the derivative of the second, plus the second times the derivative of the first."Example: Find the derivative of .The Quotient Rule:Mnemonic: "Low d-high minus high d-low, square the bottom and away we go." (d-high means derivative of the numerator).Example: Find the derivative of .6. Derivatives of Special Functions (MEMORIZE ALL)Trigonometric FunctionsAP Tip: The derivatives of the "co-" functions (cosine, cotangent, cosecant) are all negative.Exponential FunctionsThe Natural Exponential Function (most common):(It is its own derivative!)Other Bases:Example: Logarithmic FunctionsThe Natural Logarithm:Other Bases:Example: Tip: You can derive this using the change of base formula: . Then .