AP Calculus BC Unit 2: The Derivative & Fundamental Properties

Derivative

Measures the instantaneous rate of change of a function.

Average Rate of Change (AROC)

Measures how a function changes over a specific interval [a, b]; it's the slope of the secant line.

Instantaneous Rate of Change

Measures the rate of change at a specific single moment in time; it's the slope of the tangent line.

Secant Line

Line that connects two points on the graph of a function and represents the average rate of change.

Tangent Line

Line that touches the curve at exactly one point, representing the instantaneous rate of change.

Limit Definition of the Derivative

To calculate the derivative, we use limits to bring the two points of a secant line infinitesimally close together.

h Definition

Derivative is defined as f'(x) = lim_{h -> 0} [f(x+h) - f(x)] / h.

Derivative at a Point

Defined as f'(c) = lim_{x -> c} [f(x) - f(c)] / (x - c).

Lagrange Notation

Uses symbols f'(x) and y' for denoting derivatives of general functions.

Leibniz Notation

Uses symbols dy/dx and d/dx[f(x)] to show the ratio of infinitesimal changes.

Higher Order Derivative

The second derivative, denoted as f''(x) or d^2y/dx^2, represents the rate of change of the rate of change.

Estimating Derivatives from a Table

Calculate the Average Rate of Change using points surrounding the value of interest.

Estimating Derivatives from a Graph

The tangent line's slope at a point represents the derivative.

Fundamental Theorem of Differentiability

If a function is differentiable at x=c, it MUST be continuous at x=c.

Discontinuity

A break in the graph; if present, the function is not differentiable.

Corner Points

Sharp turns in the graph where the left slope does not equal the right slope.

Cusps

Extreme sharp points on a graph where the slopes approach infinity.

Vertical Tangents

Points where the slope of the tangent line is undefined.

Constant Rule

The derivative of a constant is 0.

Power Rule

To find the derivative of x^n, use d/dx[x^n] = nx^{n-1}.

Constant Multiple Rule

The derivative of a constant times a function is the constant times the derivative of the function.

Sum/Difference Rule

The derivative of a sum is the sum of the derivatives.

Natural Exponential

The derivative of e^x is e^x.

Natural Logarithm

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

Product Rule

For f(x) = u(x)v(x), use d/dx[uv] = uv' + vu'.

Quotient Rule

For f(x) = u(x)/v(x), use d/dx[u/v] = (vu' - uv')/v^2.

Trigonometric Derivative: Sine

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

Trigonometric Derivative: Cosine

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

Trigonometric Derivative: Tangent

The derivative of tan(x) is sec^2(x).

Trigonometric Derivative: Cosecant

The derivative of csc(x) is -csc(x)cot(x).

Trigonometric Derivative: Secant

The derivative of sec(x) is sec(x)tan(x).

Trigonometric Derivative: Cotangent

The derivative of cot(x) is -csc^2(x).

Common Mistakes #1: Distributing the Derivative

Distributing the derivative incorrectly leads to errors; use product or quotient rules instead.

Common Mistakes #2: Order in Quotient Rule

Always start with the denominator when applying the quotient rule.

Tangent Line Equation

The equation of the tangent line is y - f(c) = f'(c)(x - c).

Notation Errors

Ensure to include limit notation until plugging in the limit value.

Confusing Differentiability

A continuous function may not be differentiable; check for corners or cusps.

Difference between Average and Instantaneous

Average is over an interval; instantaneous is at a specific point.

Slope of a Secant Line Formula

AROC = (f(b) - f(a)) / (b - a).

Slope of a Tangent Line

The instantaneous rate of change at a point.

Approaching Limits in Definitions

Use limits to redefine the derivative when approaching zero.

Utilization of Derivative Rules

Employ derivative rules to compute derivatives efficiently.

Simplifying Derivatives

Combine basic derivative rules for more complex functions.

Analyzing Functions Graphically

Assessing steepness of curves helps indicate positive or negative slopes.

Differentiability Criteria

Continuous but non-differentiable at corners, cusps, and vertical tangents.

Concept of Smoothness in Differentiability

A differentiable function must have a 'smooth' graph without sharp turns.

Limits and Derivatives

Derivatives utilize limits to approach point evaluations.

Using Data for Derivatives

Interpolation and estimating slopes via tables provide approximate derivatives.