Unit 2: Differentiation: Definition and Fundamental Properties

Average rate of change

The slope over an interval [a,b]: (f(b)-f(a))/(b-a); corresponds to the secant line slope.

Secant line

A line that intersects a curve at two points; its slope gives the average rate of change between those points.

Slope (rise over run)

A measure of steepness: (y2-y1)/(x2-x1).

Instantaneous rate of change

The rate of change at a specific input value; equals the slope of the tangent line at that point.

Tangent line

A line that touches a curve at one point and matches the curve’s direction there; its slope is the derivative at that point.

Difference quotient

An expression like (f(a+h)-f(a))/h or (f(x)-f(a))/(x-a) used to compute average/instantaneous rates of change.

Limit

A value that an expression approaches as the input approaches a certain number (e.g., h→0).

Derivative at a point

f'(a), defined (when it exists) as a limit giving the slope of the tangent line to y=f(x) at x=a.

Definition of the derivative (h→0 form)

f'(a)=lim_{h→0} (f(a+h)-f(a))/h.

Equivalent derivative definition (x→a form)

f'(a)=lim_{x→a} (f(x)-f(a))/(x-a).

Derivative function

A function giving slope at every x where it exists: f'(x)=lim_{h→0}(f(x+h)-f(x))/h.

f'(a)

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

f'(x)

A new function whose output is the slope of f at input x (where the derivative exists).

Leibniz notation (dy/dx)

A common derivative notation meaning “the derivative of y with respect to x,” emphasizing rate of change.

Evaluation notation |_{x=a}

Notation like (dy/dx)|_{x=a}, meaning “evaluate the derivative at x=a.”

First derivative

The derivative f'(x); measures the rate of change (slope) of the original function.

Second derivative

f''(x); the derivative of f'(x), measuring how the rate of change itself is changing.

Units of a derivative

If f has output units and x has input units, then f'(x) has units (output units)/(input units).

Forward difference approximation

An estimate of f'(a): (f(a+h)-f(a))/h using a point to the right of a.

Backward difference approximation

An estimate of f'(a): (f(a)-f(a-h))/h using a point to the left of a.

Central (symmetric) difference

Often-best table estimate of f'(a): (f(a+h)-f(a-h))/(2h).

Left-hand derivative

f'-(a)=lim{h→0^-}(f(a+h)-f(a))/h; uses h approaching 0 from the negative side.

Right-hand derivative

f'+(a)=lim{h→0^+}(f(a+h)-f(a))/h; uses h approaching 0 from the positive side.

Differentiable at a point

A function is differentiable at x=a if the derivative limit exists there and is finite.

Differentiability implies continuity

If f'(a) exists, then f is continuous at a (but not vice versa).

Continuous but not differentiable

A situation where f is continuous at a point but f'(a) does not exist (e.g., at a corner of |x| at 0).

Discontinuity

A break in the function (jump, hole, or infinite behavior) that typically prevents differentiability.

Corner

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

Cusp

A sharp point where slopes become infinite in opposite directions; derivative does not exist there.

Vertical tangent

A tangent line with “infinite” slope; in AP contexts treated as not differentiable at that point.

Point-slope form

Line form using a point (x1,y1) and slope m: y-y1=m(x-x1).

Tangent line equation

At x=a: y-f(a)=f'(a)(x-a).

Constant rule

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

Power rule

For n a positive integer: d/dx(x^n)=n x^{n-1}.

Constant multiple rule

d/dx(c·g(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).

Polynomial

A function made of terms an x^n + … + a1 x + a_0; differentiated term-by-term using basic rules.

Product rule

If f(x)=u·v, then f'(x)=u·v' + v·u' (“1d2 + 2d1”).

Quotient rule

If f(x)=u/v, then f'(x)=(v·u' - u·v')/v^2.

“Low d high − high d low” (memory cue)

A reminder for the quotient rule: (denominator·(numerator)′ − numerator·(denominator)′)/(denominator)^2.

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).

Derivative of tan x

d/dx(tan x)=sec^2 x (with x in radians).

Radian measure

The angle unit assumed for standard trig derivatives; the formulas are correct when x is in radians.

Derivative of e^x

d/dx(e^x)=e^x; the function equals its own derivative.

Derivative of ln x

d/dx(ln x)=1/x (domain: x>0).

Derivative of a^x

For a>0, a≠1: d/dx(a^x)=a^x ln(a).

Marginal cost

If C(q) is cost to produce q items, then C'(q) is marginal cost in dollars per item (rate of change of cost).

Local linearity (linear approximation)

If f is differentiable at a, then near a: f(x)≈f(a)+f'(a)(x-a), using the tangent line to approximate the function.