AP Calculus AB Unit 2 Study Guide: Derivative Meaning, Limit Definition, Notation, Differentiability, and Core Rules

Derivative

A limit that gives the instantaneous rate of change of a function (equivalently, the slope of the tangent line to its graph).

Instantaneous rate of change

The rate of change at a specific input value, found by taking a limit as the interval shrinks to zero.

Average rate of change

The change in output divided by the change in input over an interval; for f from a to b it is (f(b)−f(a))/(b−a).

Slope (rise over run)

For two points (x1,y1) and (x2,y2), slope = (y2−y1)/(x2−x1).

Secant line

A line through two points on a curve; its slope represents the average rate of change over the interval between the points.

Tangent line

The line that matches the curve’s direction at a point; it is the limit of secant lines as the two points come together.

Difference quotient

The secant-slope expression (f(x+h)−f(x))/h, which approximates the derivative when h is small.

Limit definition of the derivative

f'(x) = lim(h→0) (f(x+h)−f(x))/h, defining the exact instantaneous rate of change.

Indeterminate form 0/0

The undefined result you get if you plug h=0 directly into (f(x+h)−f(x))/h; the limit process resolves this (if possible).

Left-hand derivative (one-sided)

f'_-(a) = lim(h→0−) (f(a+h)−f(a))/h, using h values approaching 0 from the left.

Right-hand derivative (one-sided)

f'_+(a) = lim(h→0+) (f(a+h)−f(a))/h, using h values approaching 0 from the right.

Differentiable at a point

A function is differentiable at x=a if f'(a) exists as a finite real number (the limit in the derivative definition exists).

Continuity

A function is continuous at x=a if there is no break there (intuitively, you can draw the graph through a without lifting your pencil).

Differentiability implies continuity

If a function is differentiable at x=a, then it must be continuous at x=a.

Contrapositive (continuity/differentiability)

If a function is not continuous at x=a, then it cannot be differentiable at x=a.

Corner

A point where the left-hand and right-hand slopes are finite but not equal, so the derivative does not exist there.

Cusp

A point where the one-sided slopes become infinite in opposite directions (e.g., +∞ from one side and −∞ from the other), so the derivative does not exist.

Vertical tangent

A point where the slope becomes infinite in the same direction from both sides; the derivative is not a finite real number.

Discontinuity

A hole, jump, or asymptote in the graph; the function is not continuous and therefore not differentiable there.

Derivative at a point (f'(a))

A single number giving the instantaneous rate of change (slope of the tangent line) at x=a.

Derivative function (f'(x))

A new function that outputs the slope of the original function at each x where the derivative exists.

Leibniz notation (dy/dx)

Notation for the derivative emphasizing the rate of change of y with respect to x.

Operator notation (d/dx)[f(x)]

Notation that treats differentiation as an operator applied to an expression f(x).

Second derivative (f''(x))

The derivative of the derivative; written f''(x) (or y''), representing how the slope (first derivative) changes.

Slope of the tangent line (geometric meaning)

At a smooth point on a curve, f'(a) equals the slope of the tangent line at x=a.

Tangent line equation

At x=a, the tangent line is y−f(a)=f'(a)(x−a).

Linearization

Using the tangent line at a point to approximate the function near that point (a local linear approximation).

Velocity (as a derivative)

If s(t) is position, then velocity is the derivative v(t)=s'(t).

Average velocity

Over t1 to t2, average velocity is (s(t2)−s(t1))/(t2−t1).

Instantaneous velocity

Velocity at a specific time t=a; given by v(a)=s'(a).

Units of a derivative

Units of (output units)/(input units), e.g., meters per second if position is meters and time is seconds.

Rationalizing (using conjugates)

An algebra technique (often for square roots) multiplying by a conjugate to simplify and cancel terms in a difference quotient.

Constant rule

d/dx[k]=0 for a constant k.

Constant multiple rule

d/dx[kf(x)]=k f'(x) for constant k.

Sum rule

d/dx[f(x)+g(x)]=f'(x)+g'(x).

Difference rule

d/dx[f(x)−g(x)]=f'(x)−g'(x).

Linearity properties (of derivatives)

Rules showing derivatives distribute over addition/subtraction and allow constants to factor out (but not over products/quotients).

Power rule

d/dx[x^n]=n x^{n−1} (multiply by the exponent and subtract 1 from the exponent).

Negative exponent (power rule application)

The power rule still applies when n is negative; e.g., d/dx[x^(−3)]=−3x^(−4)=−3/x^4.

Rational exponent (power rule application)

The power rule applies for rational exponents on the appropriate domain; e.g., d/dx[x^(1/2)]=(1/2)x^(−1/2)=1/(2√x).

Derivative of a linear function (mx+b)

d/dx[mx+b]=m, a constant slope.

Natural exponential function (e^x)

A special exponential function whose rate of change is proportional to (and equals) its value in calculus.

Derivative of e^x

d/dx[e^x]=e^x.

Natural logarithm (ln x)

The logarithm base e; in real-valued contexts it is defined for x>0.

Derivative of ln x

d/dx[ln x]=1/x (valid for x>0 in real-valued contexts).

Radian measure

Angle measurement required for standard trig derivative formulas (AP Calculus assumes radians unless stated otherwise).

Derivative of sin x

d/dx[sin x]=cos x (when x is in radians).

Derivative of cos x

d/dx[cos x]=−sin x (when x is in radians).

Product rule

d/dx[f(x)g(x)]=f'(x)g(x)+f(x)g'(x).

Quotient rule

d/dx[f(x)/g(x)]= (f'(x)g(x)−f(x)g'(x))/(g(x))^2, with g(x)≠0.