AP Calculus BC Unit 2: Derivatives as Limits, Rates of Change, and Core Differentiation Rules

Average rate of change

The average change in output per unit change in input over an interval; equals the slope of the secant line.

Secant line

A line that intersects a curve at two points; used to compute an average rate of change over an interval.

Secant slope formula

Slope between inputs a and b: (f(b)−f(a))/(b−a) (equivalently (y2−y1)/(x2−x1)).

Instantaneous rate of change

The rate of change at a single input value; equals the slope of the tangent line and is measured by the derivative.

Tangent line

A line that matches the curve’s direction locally at a point; its slope is the instantaneous rate of change (and it can cross the curve).

Derivative (concept)

A measure of instantaneous rate of change; the slope of the tangent line at a point, defined as a limit of secant slopes.

Difference quotient

An expression for secant slope using a small step: (f(x+h)−f(x))/h.

Limit definition of the derivative (h→0 form)

f′(x)=lim(h→0) (f(x+h)−f(x))/h, if the limit exists as a finite real number.

Equivalent derivative definition (x→a form)

f′(a)=lim(x→a) (f(x)−f(a))/(x−a); another way to express the same tangent-slope limit idea.

Indeterminate form 0/0 in derivatives

Plugging h=0 into (f(x+h)−f(x))/h gives (f(x)−f(x))/0 = 0/0 (undefined); you must simplify first, then take the limit.

Prime notation

A common notation for the derivative: f′(x) (and f″(x) for the second derivative).

Leibniz notation

Derivative notation emphasizing rate/units: dy/dx; in this unit, treat it as one symbol meaning “the derivative,” not as an ordinary fraction to cancel.

Operator notation

Derivative written as an operator acting on a function: d/dx [f(x)].

Second derivative

The derivative of the derivative (rate of change of the rate of change): f″(x) or y″.

Derivative at a point vs. derivative function

f′(a) is a single number (slope at x=a); f′(x) is a new function giving the slope for each x where it exists.

Estimating a derivative from a graph

Approximate f′(a) by sketching/imagining the tangent line at x=a, choosing two convenient points on that tangent line, and computing rise/run.

Estimating a derivative from a table

Approximate f′(a) using average rates of change over small intervals near a (difference quotients from nearby table values).

One-sided difference quotient estimate

A one-sided approximation: f′(a)≈(f(a+h)−f(a))/h using a small h from one side of a.

Centered difference quotient estimate

Often a better table-based estimate: f′(a)≈(f(a+h)−f(a−h))/(2h).

Local linearity

If a function is differentiable at a, then near a its graph looks almost like a line (the tangent line), enabling linear approximation.

Linear approximation (tangent line approximation)

Near x=a: f(x)≈f(a)+f′(a)(x−a).

Differential approximation

For small changes: Δf≈f′(a)Δx (change in output ≈ derivative × change in input).

Units of the derivative

Derivative units are “output units per input unit” (e.g., meters/second if output is meters and input is seconds).

Marginal cost

An interpretation of a derivative in economics: if C(q) is cost, then C′(q) is the approximate cost increase (dollars per item) for producing one more item near q.

Sign of the derivative

f′(x)>0 means increasing; f′(x)<0 means decreasing; f′(x)=0 means locally flat (horizontal tangent).

Tangent line crossing misconception

In calculus, “tangent” means matching slope locally, not “touching once and never crossing”; tangent lines can cross the curve.

Function value vs. derivative value mistake

f(a) is a height/output value; f′(a) is a slope/rate of change. Confusing them leads to wrong interpretations and tangent lines.

Differentiable at a point

A function is differentiable at a if lim(h→0) (f(a+h)−f(a))/h exists as a finite real number (so a well-defined tangent slope exists).

Differentiability implies continuity

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

Continuous but not differentiable

A function can be continuous yet fail to have a derivative (e.g., at corners, cusps, or vertical tangents).

Corner

A point where left and right slopes are finite but unequal, so the derivative does not exist (classic example: |x| at 0).

Cusp

A sharp point where slopes become infinite in opposite directions; the derivative does not exist as a finite real number.

Vertical tangent

A point where the tangent line is vertical, corresponding to an infinite slope; the derivative does not exist as a finite real number.

Discontinuity (effect on differentiability)

Any discontinuity (jump, hole, asymptote) prevents differentiability at that input because differentiability requires continuity.

Endpoint derivative issue

At an endpoint of a domain, a two-sided derivative may not exist; sometimes only a one-sided derivative can be defined.

Right-hand derivative

f′+(a)=lim(h→0+) (f(a+h)−f(a))/h (uses values to the right of a).

Left-hand derivative

f′−(a)=lim(h→0−) (f(a+h)−f(a))/h (uses values to the left of a).

Constant rule

The derivative of a constant is zero: d/dx[c]=0.

Constant multiple rule

Constants factor out of derivatives: d/dx[c·f(x)]=c·f′(x).

Power rule (integer powers)

For n a nonnegative integer: d/dx[x^n]=n·x^(n−1) (“multiply down and decrease the power”).

Linearity (sum/difference rule)

Derivatives distribute over addition/subtraction: d/dx[f(x)±g(x)]=f′(x)±g′(x).

Tangent line equation (point-slope form)

Tangent line at x=a: y−f(a)=f′(a)(x−a).

Product rule

Derivative of a product: d/dx[f(x)g(x)]=f′(x)g(x)+f(x)g′(x) (not the product of derivatives).

Quotient rule

Derivative of a quotient: d/dx[f(x)/g(x)]= (f′(x)g(x)−f(x)g′(x))/(g(x))^2, with g(x)≠0.

Derivative of e^x

d/dx[e^x]=e^x (the natural exponential is its own derivative).

Derivative of a^x

For a>0, a≠1: d/dx[a^x]=a^x·ln(a).

Derivative of ln(x)

d/dx[ln(x)]=1/x (for x>0).

Derivative of sin(x)

d/dx[sin(x)]=cos(x) (standard formula assumes x is in radians).

Derivative of cos(x)

d/dx[cos(x)]=−sin(x) (watch the negative sign; x must be in radians).

Other trig derivatives (radians)

d/dx[tan(x)]=sec^2(x); d/dx[sec(x)]=sec(x)tan(x); d/dx[csc(x)]=−csc(x)cot(x); d/dx[cot(x)]=−csc^2(x) (inputs in radians).