Unit 2: Differentiation: Definition and Fundamental Properties

Average rate of change

The change in a function’s output over an interval divided by the change in input: (f(b)−f(a))/(b−a).

Secant line

A line that intersects a curve at two points; used to model average change between those points.

Secant slope

The slope of a secant line through (a,f(a)) and (b,f(b)); equals the average rate of change over [a,b].

Instantaneous rate of change

The rate of change at a single input value; geometrically, the slope of the tangent line at that point.

Tangent line

A line that touches a curve at a point and matches the curve’s direction there (when it exists).

Tangent slope

The slope of the tangent line at x=a; this is the value of the derivative f'(a) if it exists.

Increment h

A small change in input used to compare x=a to x=a+h when forming secant slopes near a point.

Difference quotient

The expression (f(a+h)−f(a))/h, representing change in f divided by change in x.

Limit as h → 0

The process used to turn a secant slope (difference quotient) into a tangent slope by letting the interval shrink toward zero.

Derivative at a (f'(a))

Defined by f'(a)=lim_{h→0}(f(a+h)−f(a))/h, if the limit exists as a finite real number.

Differentiable at a

A function is differentiable at x=a if the derivative limit exists (finite) at that point.

Equivalent limit form (x → a)

An alternate definition: f'(a)=lim_{x→a}(f(x)−f(a))/(x−a).

Derivative as a limit of slopes

The derivative is the limiting value of secant slopes as the second point approaches the point of tangency.

Left-hand derivative (f'_-(a))

The derivative estimated/defined by approaching from the left: lim_{h→0^-}(f(a+h)−f(a))/h.

Right-hand derivative (f'_+(a))

The derivative estimated/defined by approaching from the right: lim_{h→0^+}(f(a+h)−f(a))/h.

One-sided derivative test

f'(a) exists iff both f'-(a) and f'+(a) exist and are equal.

Differentiability implies continuity

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

Continuity does not imply differentiability

A function can be continuous but still not have a derivative at a point (e.g., at corners or cusps).

Discontinuity

A break in the graph (hole, jump, or asymptote); prevents the derivative from existing at that point.

Corner

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

Cusp

A sharp point where slopes become infinite in opposite directions (or behave vertically), so no single tangent slope exists.

Vertical tangent

A tangent line that is vertical; the slope is undefined, so the derivative at that point does not exist as a finite number.

Rationalizing (conjugate technique)

An algebra method (often for radicals) that multiplies by a conjugate to simplify a difference quotient so the limit can be evaluated.

Conjugate

For √(a+h)−√a, the conjugate is √(a+h)+√a; multiplying by it creates a difference of squares.

Derivative function (f'(x))

A new function giving the derivative (tangent slope) at each x where the original function is differentiable: f'(x)=lim_{h→0}(f(x+h)−f(x))/h.

Leibniz notation (dy/dx)

Notation for the derivative meaning “rate of change of y with respect to x,” especially useful in applied rate problems.

Derivative operator (d/dx)

Notation emphasizing differentiation as an operation, e.g., d/dx[f(x)].

Second derivative (f''(x))

The derivative of the derivative; written f''(x) or y''.

Units of derivative

If y=f(x), then f'(a) has units of “units of y per unit of x.”

Average velocity

For position s(t), average velocity on [a,b] is (s(b)−s(a))/(b−a).

Instantaneous velocity

For position s(t), instantaneous velocity at t=a is s'(a).

Tangent line equation

If f is differentiable at x=a, the tangent line is y−f(a)=f'(a)(x−a).

Normal line

The line perpendicular to the tangent line at x=a, passing through (a,f(a)).

Negative reciprocal (normal slope rule)

If the tangent slope is m=f'(a) (and m≠0), the normal slope is −1/m.

Horizontal tangent

A tangent line with slope 0; occurs where f'(a)=0 (if the derivative exists).

Constant rule

If f(x)=c (a constant), then f'(x)=0.

Constant multiple rule

If g(x)=c·f(x), then g'(x)=c·f'(x).

Sum rule

If h(x)=f(x)+g(x), then h'(x)=f'(x)+g'(x).

Difference rule

If h(x)=f(x)−g(x), then h'(x)=f'(x)−g'(x).

Power rule

d/dx[x^n]=n·x^{n−1} (commonly used for polynomials and rewritten powers).

Rewriting with negative exponents

Converting forms like 1/x^3 to x^{−3} to apply the power rule more directly.

Product rule

If f(x)=u·v, then f'(x)=u·(dv/dx)+v·(du/dx).

Quotient rule

If f(x)=u/v, then f'(x)=(v·du/dx − u·dv/dx)/v^2.

Symmetric difference quotient

A often-more-accurate table estimate: f'(a)≈(f(a+h)−f(a−h))/(2h).

Forward difference quotient

A right-hand estimate from data: f'(a)≈(f(a+h)−f(a))/h.

Derivative of sin(x)

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

Derivative of cos(x)

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

Radians (for trig derivatives)

The formulas for derivatives of sin and cos are valid when angles are measured in radians; degrees require a conversion factor.

Derivative of e^x

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

Derivative of ln(x)

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