Differentiation: Definition and Fundamental Properties

Rate of change

A measure of how an output quantity changes in response to changes in an input quantity (how sensitive f(x) is to changes in x).

Average rate of change

Change over an interval: (f(b)−f(a))⁄(b−a); the slope between two points on the graph.

Instantaneous rate of change

The rate of change at a single input value; the slope at a point on the curve (formalized by the derivative).

Secant line

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

Tangent line

The line that touches a curve at a point and matches its local direction; its slope is the derivative (when it exists).

Slope (rise over run)

A ratio measuring steepness: (change in y)⁄(change in x); in calculus it can be taken at a single point via limits.

Difference quotient

An average-rate expression used to define derivatives, e.g., (f(a+h)−f(a))⁄h or (f(x)−f(a))⁄(x−a).

Limit (h→0)

The process of letting h shrink toward 0 to capture a value approached by secant slopes, producing the tangent slope.

Derivative

f′(a) measures instantaneous rate of change at x=a; geometrically, the slope of the tangent line at x=a.

Derivative at a point (f′(a))

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

Derivative function (f′(x))

A new function that outputs the slope of the original function at each x where the function is differentiable.

Limit definition of the derivative (h-form)

f′(a)=lim(h→0) (f(a+h)−f(a))⁄h (if the limit exists and is finite).

Limit definition of the derivative (x→a form)

f′(a)=lim(x→a) (f(x)−f(a))⁄(x−a); equivalent to the h-form definition.

One-sided derivative

A derivative computed from only one side of a point; the derivative exists only if left and right derivatives both exist and match.

Left-hand derivative

lim(h→0−) (f(a+h)−f(a))⁄h; the slope approached from inputs less than a.

Right-hand derivative

lim(h→0+) (f(a+h)−f(a))⁄h; the slope approached from inputs greater than a.

Differentiable (at x=a)

A function is differentiable at a if f′(a) exists (the derivative limit exists and is finite).

Continuous (at x=a)

A function is continuous at a if lim(x→a) f(x)=f(a) (no break in the graph at that point).

Differentiability implies continuity

If f is differentiable at a, then f must be continuous at a (but continuity does not guarantee differentiability).

Corner

A point where the function is continuous but the left-hand and right-hand slopes are finite and unequal, so f′(a) does not exist.

Cusp

A sharp point where slopes become unbounded with opposite behavior from left and right; derivative does not exist as a finite number.

Vertical tangent

A point where the tangent line is vertical and the slope tends to ±∞; the derivative is not a finite real number there.

Discontinuity

A break/jump/hole in the graph; if a function is not continuous at a point, it cannot be differentiable there.

Leibniz notation (dy/dx)

A common derivative notation emphasizing rate of change (“change in y per change in x”), useful for applied interpretation.

Operator notation (d/dx)

Notation for “differentiate with respect to x,” e.g., (d/dx)(f(x))=f′(x).

Second derivative (f′′(x))

The derivative of the derivative; measures how the slope (rate of change) itself changes with x.

Units of a derivative

Derivative units are “output units per input unit,” e.g., meters per second if position is meters and time is seconds.

Velocity

If s(t) is position, velocity is v(t)=s′(t); it is the instantaneous rate of change of position.

Speed

The magnitude of velocity: |v(t)|=|s′(t)|; always nonnegative.

Acceleration

The rate of change of velocity: a(t)=v′(t)=s′′(t) (conceptually, how velocity changes over time).

Marginal cost

If C(x) is cost to produce x items, C′(x) is marginal cost (approximately dollars per additional item).

Local linearity

The idea that near a differentiable point, a function behaves approximately like its tangent line (looks nearly linear when zoomed in).

Linear approximation (tangent-line approximation)

Using f(a)+f′(a)(x−a) to estimate f(x) for x near a; based on the tangent line at x=a.

Point-slope form (tangent line)

Equation of a line with slope m through (a,f(a)): y−f(a)=m(x−a); for tangents, m=f′(a).

Constant function rule

If f(x)=c (constant), then f′(x)=0 because the output does not change as x changes.

Power rule

d/dx(x^n)=n x^(n−1) (for appropriate domains); “multiply down and decrease the power.”

Constant multiple rule

If f(x)=c·g(x), then f′(x)=c·g′(x) (constants factor out of derivatives).

Sum rule

d/dx(g(x)+h(x))=g′(x)+h′(x).

Difference rule

d/dx(g(x)−h(x))=g′(x)−h′(x).

Product rule

If f(x)=u·v, then f′(x)=u·v′+v·u′ (not u′·v′).

Quotient rule

If f(x)=u/v, then f′(x)=(v·u′−u·v′)/v^2.

“Low d high − high d low”

Mnemonic for the quotient rule numerator: denominator·(derivative of numerator) minus numerator·(derivative of denominator), all over denominator squared.

Derivative of sin x

d/dx(sin x)=cos x (angles must be in radians).

Derivative of cos x

d/dx(cos x)=−sin x (angles must be in radians; note the negative sign).

Radians requirement (trig derivatives)

The standard derivative formulas for sin and cos work correctly only when angles are measured in radians.

Derivative of e^x

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

Derivative of ln x

d/dx(ln x)=1/x (for x>0 in real-valued calculus).

Domain restriction (log/reciprocal)

ln x requires x>0; the derivative 1/x requires x≠0; always consider domain when differentiating.

Secant-slope estimate from a table/graph

Approximating f′(a) by computing nearby average rates: (f(a+h)−f(a))/h from small positive and negative h values.

Smooth join (piecewise differentiability)

At a piecewise junction, a “smooth” connection requires continuity (matching values) and differentiability (matching left/right slopes).