AP Calculus AB Unit 2 Study Guide: Derivatives by Definition, Interpretation, and Core Rules

Derivative

A mathematical tool that gives the instantaneous rate of change (slope of the tangent line) at a specific input.

Secant Line

A line that intersects a curve at two or more points, representing the average rate of change over an interval.

Tangent Line

A line that touches a curve at a single point and represents the instantaneous rate of change.

Limit Definition of Derivative

The derivative of a function at a point is defined as f'(a) = lim(h -> 0) [f(a+h) - f(a)] / h.

Average Rate of Change (AROC)

The average rate of change of a function f over an interval [a,b] is given by (f(b) - f(a)) / (b - a).

Instantaneous Rate of Change

The rate at which a function is changing at a specific point, represented by the derivative.

Differentiable

A function is said to be differentiable at a point if its derivative exists at that point.

Continuous Function

A function is continuous at a point if the limit as it approaches the point equals the function's value at that point.

Power Rule

The derivative of x^n is n*x^(n-1).

Product Rule

If f(x) = u(x)v(x), then f'(x) = u'v + uv'.

Quotient Rule

If f(x) = u(x) / v(x), then f'(x) = (v(x)u'(x) - u(x)v'(x)) / (v(x))^2.

Derivative of Sin(x)

The derivative of sin(x) is cos(x).

Derivative of Cos(x)

The derivative of cos(x) is -sin(x).

Exponential Function Derivative

The derivative of e^x is e^x.

Natural Log Derivative

The derivative of ln(x) is 1/x.

Change in Output

The difference in the values of a function, typically represented as f(x+h) - f(x).

Change in Input

The difference in input values, commonly represented as h.

Tangent Slope at a Point

The slope of the tangent line to the curve at a specific point, representing the instantaneous rate of change of the function.

Definition of Continuity

A function is continuous at a point x=a if f(a) is defined, lim(x -> a) f(x) exists, and lim(x -> a) f(x) = f(a).

One-Sided Derivative

The derivative of a function from one side only: right-hand derivative limits as h approaches 0 from the positive side and left-hand derivative limits as h approaches 0 from the negative side.

Local Linearity

Near a point where a function is differentiable, the graph looks almost like a straight line.

Linear Approximation

Using the tangent line at a point to approximate the value of a function near that point.

Cusp

A point where a function has a sharp point and may fail to be differentiable.

Corner

A point on the graph where the derivative does not exist due to differing slopes from either side.

Vertical Tangent

A point where the derivative is undefined because slope approaches infinity.

Chain Rule

If f(g(x)) is a composite function, then the derivative is f'(g(x)) * g'(x).

Instantaneous Change

The rate of change of a quantity at a specific instant, often identified using derivatives.

Derivative Notation

Common notations for the derivative include f'(x), dy/dx, and d/dx[f(x)].

Rate of Change

The change in one quantity relative to the change in another, often measured using derivatives.

Average Velocity

The average rate of change of position with respect to time over a given interval.

Supplementary Function

A function that is derived from the initial function, often related through changes in its inputs.

Tangential Velocity

The instantaneous velocity of an object moving along a curved path at a specific point.

Differentiation Process

The method of determining the derivative of a function, often using limits.

Tangent Line Approximation

Using the derivative at a point to estimate the value of the function close to that point.

Higher Derivatives

Derivatives of derivatives, providing information about the curvature and changing rates of change of a function.

Slope of a Function

A measure of how steeply a function rises or falls at a given point; the derivative gives this value.

Rate of Change of a Function

The value of the derivative at a particular point, indicating how the function value is changing.

Graphical Interpretation

Visual representation of functions, including their values and slopes as they relate to derivatives.

Derivative at a Point

The specific value of the derivative function at a particular input.

Critical Point

A point on the graph of a function where the derivative is zero or undefined, potentially indicating local maxima or minima.

Symmetry in Functions

Behavior of functions where changes in inputs correspond to consistent outputs or derivatives.

Limit Existence at a Point

A condition required for a function's derivative to be defined at a specific point.